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Sample Size Selection and Hypothesis Tests Lecture 16

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Sample Size Selection and Hypothesis Tests Lecture 16

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    1. Sample Size Selection and Hypothesis Tests Lecture 16 William F. Hunt, Jr. Statistics 361 Section 7.5 & 8.1

    2. The Basic Paradigm. Recall that the basic paradigm of statistics is that we have a population of interest from which we take a sample. From that sample we calculate a sample statistic and use the statistic to make inference to the population parameter. Recall that the basic paradigm of statistics is that we have a population of interest from which we take a sample. From that sample we calculate a sample statistic and use the statistic to make inference to the population parameter.

    3. Statistic ± multiplier*standard error Confidence Interval Recall from the previous section that we can calculate the margin of error for either the proportion or a mean. Recall that we use proportions when we are asking our subjects a yes/no question and we use the mean when we are asking a numeric question. Both these formulas have the form statistic plus or minus the margin of error. The margin of error is constructed by multiplying a multiplier times the standard error.Recall from the previous section that we can calculate the margin of error for either the proportion or a mean. Recall that we use proportions when we are asking our subjects a yes/no question and we use the mean when we are asking a numeric question. Both these formulas have the form statistic plus or minus the margin of error. The margin of error is constructed by multiplying a multiplier times the standard error.

    4. Choosing a Sample Size Margin of Error Depends on Sample Size In designing an experiment or study we need to decide how large n should be. Precision- acceptable distance between statistic and parameter. Largest acceptable MOE Bound on MOE => B. The margin of error in both these formulas depend on the sample size. In setting up an experiment we need to determine the size of the sample. In this section we will examine how we can determine the appropriate sample size. To determine an appropriate sample size we must first determine how large of a margin of error will be acceptable. The largest acceptable margin of error is refered to as the precision. So for example we might want to estimate the average income of home owners in Wake County NC to within plus or minus $10,000 . This is the bound on the margin of error. We will use the notation B for the bound on the margin of error. Prior to determining the sample size we will need to specify the bound B. When we have specified the Bound we can use the formulas for a margin of error to determine the appropriate sample size. Lets begin by examining the formula for the margin of error. The margin of error in both these formulas depend on the sample size. In setting up an experiment we need to determine the size of the sample. In this section we will examine how we can determine the appropriate sample size. To determine an appropriate sample size we must first determine how large of a margin of error will be acceptable. The largest acceptable margin of error is refered to as the precision. So for example we might want to estimate the average income of home owners in Wake County NC to within plus or minus $10,000 . This is the bound on the margin of error. We will use the notation B for the bound on the margin of error. Prior to determining the sample size we will need to specify the bound B. When we have specified the Bound we can use the formulas for a margin of error to determine the appropriate sample size. Lets begin by examining the formula for the margin of error.

    5. For Means Use the margin of error formula to solve for the sample size. Recall that for the mean we can write the margin of error formula as Z times s over the square root of the sample size. Recall that for the mean we can write the margin of error formula as Z times s over the square root of the sample size.

    6. For Means Use the margin of error formula to solve for the sample size. We will determine the appropriate sample size by putting in our value for B the bound on the margin of error into the formula. So in other words if we want the MOE to be no more than plus or minus 10 then the Z times s over square root of n will be equal to $10000. In this setting we will solve for the appropriate sample size. Before we examine that calculation lets take up one point about the margin of error. We will determine the appropriate sample size by putting in our value for B the bound on the margin of error into the formula. So in other words if we want the MOE to be no more than plus or minus 10 then the Z times s over square root of n will be equal to $10000. In this setting we will solve for the appropriate sample size. Before we examine that calculation lets take up one point about the margin of error.

    7. For Means Use the margin of error formula to solve for the sample size. Recall that we use the sample standard deviation in this formula because we don’t know the population standard deviation. In other words we use s because we don’t know sigma. So in reality we would like to plug in the value of sigma if we have one. So for our calculations we will use sigma and discuss how we can obtain a guess at that quantity.Recall that we use the sample standard deviation in this formula because we don’t know the population standard deviation. In other words we use s because we don’t know sigma. So in reality we would like to plug in the value of sigma if we have one. So for our calculations we will use sigma and discuss how we can obtain a guess at that quantity.

    8. For means So in our formula we will use the symbol sigma for the standard deviation. Now we need to solve this formula for the value of n. So in our formula we will use the symbol sigma for the standard deviation. Now we need to solve this formula for the value of n.

    10. For means We can rearrange terms and get the formula given by your text. Lets take a look at the components of this formula. We can rearrange terms and get the formula given by your text. Lets take a look at the components of this formula.

    11. For means The Z-score in this formula is the confidence coefficient. This value is determined by the confidence level we want. For instance if we want 95% confidence then we will use 1.96. If we want 90% confidence we would use 1.64 and so on. The Z-score in this formula is the confidence coefficient. This value is determined by the confidence level we want. For instance if we want 95% confidence then we will use 1.96. If we want 90% confidence we would use 1.64 and so on.

    12. For means The value of B is the bound on the MOE. Again the bound is the largest acceptable margin of error. We must determine this by considering what value we will accept. So if the goal of our study is to estimate the value to within plus or minus 1000 dollars or plus or minus 3 feet we will set the value of B to that amount.The value of B is the bound on the MOE. Again the bound is the largest acceptable margin of error. We must determine this by considering what value we will accept. So if the goal of our study is to estimate the value to within plus or minus 1000 dollars or plus or minus 3 feet we will set the value of B to that amount.

    13. For means The value of sigma squared would be the variance of the quantity we are interested in. We need to determine how to estimate the value of sigma. There are several options to determine the appropriate value of sigma.The value of sigma squared would be the variance of the quantity we are interested in. We need to determine how to estimate the value of sigma. There are several options to determine the appropriate value of sigma.

    14. Where do we get ?2? Estimate of Variance Pilot studies Other similar studies Expert Knowledge To get the appropriate value of sigma we have several options. One is to conduct a smaller study that will allow us to estimate the variance of the population. For instance we might have a sample of only 10 or 20 items that are used to estimate the variance. We might also determine an estimate of the variance by using other studies that are examining something similar. So for instance if we are estimating the income in Wake County we might use an estimate from a similar county elsewhere. A final method would be to use expert knowledge of someone who is an expert on the item you are examining. An expert on the particular topic may be able to estimate the variability in the variable we are examining. Lets take an example and see how we can use this formula.To get the appropriate value of sigma we have several options. One is to conduct a smaller study that will allow us to estimate the variance of the population. For instance we might have a sample of only 10 or 20 items that are used to estimate the variance. We might also determine an estimate of the variance by using other studies that are examining something similar. So for instance if we are estimating the income in Wake County we might use an estimate from a similar county elsewhere. A final method would be to use expert knowledge of someone who is an expert on the item you are examining. An expert on the particular topic may be able to estimate the variability in the variable we are examining. Lets take an example and see how we can use this formula.

    16. Note: We will always round up to the next integer on sample sizes, regardless of the amount of the fraction. Even 150.03 would round up to 151. This brings up an important note. When doing sample size calculations always round up to the next integer. Thus even a sample of 150.03 would round up to 151. This brings up an important note. When doing sample size calculations always round up to the next integer. Thus even a sample of 150.03 would round up to 151.

    17. For Proportions Lets start by looking at the formula for the margin of error for a proportion. We can specify the largest margin of error we would accept which means the largest this quantity could be. We can again solve this formula for n. Lets start by looking at the formula for the margin of error for a proportion. We can specify the largest margin of error we would accept which means the largest this quantity could be. We can again solve this formula for n.

    18. For Proportions Using some high school algebra we would find that n is given by this formula.Using some high school algebra we would find that n is given by this formula.

    19. For Proportions Again this formula includes the confidence coefficient Z. This is determined by the confidence level we want. So if we want 99% we would use 2.58 95% 1.96 and so on. Again this formula includes the confidence coefficient Z. This is determined by the confidence level we want. So if we want 99% we would use 2.58 95% 1.96 and so on.

    20. For Proportions This formula also includes the maximum margin of error we would accept. Here we are talking about proportions so this quantity will be a number between 0 and 1. For instance we might want a margin of error of 2%. For this we would use 0.02 for B. This formula also includes the maximum margin of error we would accept. Here we are talking about proportions so this quantity will be a number between 0 and 1. For instance we might want a margin of error of 2%. For this we would use 0.02 for B.

    21. For Proportions The other thing we need is a proportion. Again the point of this margin of error is to tell us what that population proportion is. We might estimate it using the sample but again the point of this formula is to determine how large the sample needs to be. So just as with the variance in the case of the mean we have several options to find pi.The other thing we need is a proportion. Again the point of this margin of error is to tell us what that population proportion is. We might estimate it using the sample but again the point of this formula is to determine how large the sample needs to be. So just as with the variance in the case of the mean we have several options to find pi.

    22. Estimate of ? Use pilot studies, similar studies etc. Use safe value What value of p makes the sample size largest? We can again use pilot studies and other studies of similar subjects that are in the literature. However we have another option. Proportions are bounced between 0 and 1. We might look for a value of pi that makes the sample size the largest.We can again use pilot studies and other studies of similar subjects that are in the literature. However we have another option. Proportions are bounced between 0 and 1. We might look for a value of pi that makes the sample size the largest.

    23. For Proportions If we look back to the formula we notice that if the pi times 1 minus pi term is large the sample size will be large. So lets play around with different values of pi to see if we can find one that makes this term as large as possible.If we look back to the formula we notice that if the pi times 1 minus pi term is large the sample size will be large. So lets play around with different values of pi to see if we can find one that makes this term as large as possible.

    24. Estimate of ? Use pilot studies, similar studies etc. Use safe value What value of p makes the sample size largest? So lets set up a table that has different values of pi and then calculate the pi times 1 minus pi. So lets set up a table that has different values of pi and then calculate the pi times 1 minus pi.

    25. Estimate of ? Use pilot studies, similar studies etc. Use safe value What value of p makes the sample size largest? We can begin with something big like 0.8. When pi is 0.8, 1 minus pi is 0.2. multiplying the two gives us.We can begin with something big like 0.8. When pi is 0.8, 1 minus pi is 0.2. multiplying the two gives us.

    26. Estimate of ? Use pilot studies, similar studies etc. Use safe value What value of p makes the sample size largest? 0.16. now lets try some other values. Maybe something smaller 0.16. now lets try some other values. Maybe something smaller

    27. Estimate of ? Use pilot studies, similar studies etc. Use safe value What value of p makes the sample size largest? Perhaps 0.6. that makes 1 minus pi 0.4. multiplying them together gives:Perhaps 0.6. that makes 1 minus pi 0.4. multiplying them together gives:

    28. Estimate of ? Use pilot studies, similar studies etc. Use safe value What value of p makes the sample size largest? 0.24 So making pi smaller actually increased pi times 1 minus pi. Lets try another smaller term. 0.24 So making pi smaller actually increased pi times 1 minus pi. Lets try another smaller term.

    29. Estimate of ? Use pilot studies, similar studies etc. Use safe value What value of p makes the sample size largest? 0.5 gives 1 minus pi of 0.5 as well. Multiplying them gives0.5 gives 1 minus pi of 0.5 as well. Multiplying them gives

    30. Estimate of ? Use pilot studies, similar studies etc. Use safe value What value of p makes the sample size largest? 0.25. Lets try some other values.0.25. Lets try some other values.

    31. Estimate of ? Use pilot studies, similar studies etc. Use safe value What value of p makes the sample size largest? 0.3 gives 0.7 for 1 minus pi. 0.3 gives 0.7 for 1 minus pi.

    32. Estimate of ? Use pilot studies, similar studies etc. Use safe value What value of p makes the sample size largest? Which gives 0.21 for a product. So now decreasing pi has made the product smaller. Which gives 0.21 for a product. So now decreasing pi has made the product smaller.

    33. Estimate of ? Use pilot studies, similar studies etc. Use safe value What value of p makes the sample size largest? Trying 0.1 gives us 0.9 as the difference and a product of Trying 0.1 gives us 0.9 as the difference and a product of

    34. Estimate of ? Use pilot studies, similar studies etc. Use safe value What value of p makes the sample size largest? 0.09. so the product is greatest at 0.5. So what is the point. If we don’t have any idea about pi, and we can’t do a pilot study or get an estimate from someplace we can use the “safe” value. Using 0.5 for pi will make the sample size as large as possible. That may cost us more but we will be assured of obtaining the margin of error we want.0.09. so the product is greatest at 0.5. So what is the point. If we don’t have any idea about pi, and we can’t do a pilot study or get an estimate from someplace we can use the “safe” value. Using 0.5 for pi will make the sample size as large as possible. That may cost us more but we will be assured of obtaining the margin of error we want.

    35. Class Problem Calculate the sample size needed to estimate p with 95% confidence using a B value of .1, the largest acceptable margin of error. Assume that you do not have a good estimate of . HINT: use the maximum estimate of

    36. Class Problem Calculate the sample size needed to estimate p with 95% confidence using a B value of .1, the largest acceptable margin of error. Assume that you do not have a good estimate of . HINT: use the maximum estimate of Discuss polling B = MOE = .03 How big a sample should you take? Discuss polling B = MOE = .03 How big a sample should you take?

    38. Other Considerations Multiple parameters to estimate Estimate both income and proportion of registered voters in Boone county. Use the larger of the two estimates. Limits on resources Money is often a driving factor A couple of quick notes about sample size determinations. In many studies we may be estimating multiple parameters. That means we will need to do many calculations such as we just did. We will need to use the sample size that is the largest. The other major issue in sample size determination is money. Often studies are limited by the amount of resources available to take the sample. A researcher may only have enough money to sample 50 people even though the formula may dictate 150 people. In those situations the researcher may be forced to accept a larger margin of error.A couple of quick notes about sample size determinations. In many studies we may be estimating multiple parameters. That means we will need to do many calculations such as we just did. We will need to use the sample size that is the largest. The other major issue in sample size determination is money. Often studies are limited by the amount of resources available to take the sample. A researcher may only have enough money to sample 50 people even though the formula may dictate 150 people. In those situations the researcher may be forced to accept a larger margin of error.

    39. Psychic? Are you psychic? One method that has been used to evaluate psychic ability is to test the subjects with Zener cards. Zener cards have one of five shapes on them. Lets begin with an example: Are you psychic. Can you predict the future? Can you guess a playing card before it is turned over? A classic experiment that is carried out to evaluate psychic ability involves Zener Cards. Zener cards have one of five shapes on them. The cards use shapes rather than the typical numbers and colors that are on regular playing cards. Shapes may be able to be more easily conveyed using mental images.Lets begin with an example: Are you psychic. Can you predict the future? Can you guess a playing card before it is turned over? A classic experiment that is carried out to evaluate psychic ability involves Zener Cards. Zener cards have one of five shapes on them. The cards use shapes rather than the typical numbers and colors that are on regular playing cards. Shapes may be able to be more easily conveyed using mental images.

    40. Psychic experiment Subject was presented with 100 cards. For each they have a 0.2 chance of getting the answer correct. If a subject is just guessing, what distribution would we expect for the proportion the subject gets correct? Lets take a look at some specifics. In a particular experiment a subject was presented with 100 Zener cards. For each one they have a 0.2 chance of getting the answer correct. If the subject is just guessing we would expect them to get about 20% of the questions correct. But the proportion will vary depending on the particular sample. What will the distribution of the proportion look like? Lets take a look at some specifics. In a particular experiment a subject was presented with 100 Zener cards. For each one they have a 0.2 chance of getting the answer correct. If the subject is just guessing we would expect them to get about 20% of the questions correct. But the proportion will vary depending on the particular sample. What will the distribution of the proportion look like?

    41. Psychic experiment Normal distribution Centered around true proportion If just guessing 0.2 Standard deviation Remember that if we have a large random sample the sample proportion should have a normal distribution centered around the true population proportion. In this case, if the person is just guessing that distribution should be centered around 0.2 and have a standard deviation of 0.04Remember that if we have a large random sample the sample proportion should have a normal distribution centered around the true population proportion. In this case, if the person is just guessing that distribution should be centered around 0.2 and have a standard deviation of 0.04

    42. Does a person have psychic powers? Decision process. Might be just guessing => should get around 20% correct. Might have psychic ability => should get significantly more than 20% correct. Could they get a lot correct just by chance? Need to use the distribution We can outline a decision process based on what could happen. The person might be just guessing, if so they should get around 20% correct. If on the other hand the person has psychic powers then they should get significantly more than 20% correct. But how much more? If they were just guessing they might get more than 20% correct but how much more? This is a place where we could use the distribution of a proportion to help our decision. We can outline a decision process based on what could happen. The person might be just guessing, if so they should get around 20% correct. If on the other hand the person has psychic powers then they should get significantly more than 20% correct. But how much more? If they were just guessing they might get more than 20% correct but how much more? This is a place where we could use the distribution of a proportion to help our decision.

    43. Starting Idea Skeptical idea: Subject does not have any telepathy powers and they are just guessing. Alternative idea: Subject has telepathy and can correctly guess the cards more than would be considered likely by chance. In outlining our decision process we can consider two basic ideas. The first is the skeptical idea: that the subject is just guessing. The alternative idea is that the subject has telepathy. The key will be to use what we know about the sampling distribution to try to decide between these two. In outlining our decision process we can consider two basic ideas. The first is the skeptical idea: that the subject is just guessing. The alternative idea is that the subject has telepathy. The key will be to use what we know about the sampling distribution to try to decide between these two.

    44. 44 Terminology Null Hypothesis – beginning claim. Status Quo idea=> Nothing new Usually that parameter is equal to specific amount Allows establishment of sampling distribution Notation: H0 “H-naught” e.g. H0 : p= 0.20 In science the skeptical idea we had is what we refer to as a null hypothesis. A null hypothesis is a status quo idea that nothing is new. Nothing is different from what we had before. This idea is often translated into a statistical question by specifying that a parameter is equal to a specific amount. By specifying an idea about a parameter we can establish a sampling distribution for the statistic. We use the notation h-naught to indicate the null hypothesis. For instance if the subject is just guessing then the proportion he should get correct would be around 0.20. In that case pi would be 0.2 In science the skeptical idea we had is what we refer to as a null hypothesis. A null hypothesis is a status quo idea that nothing is new. Nothing is different from what we had before. This idea is often translated into a statistical question by specifying that a parameter is equal to a specific amount. By specifying an idea about a parameter we can establish a sampling distribution for the statistic. We use the notation h-naught to indicate the null hypothesis. For instance if the subject is just guessing then the proportion he should get correct would be around 0.20. In that case pi would be 0.2

    45. 45 Terminology Alternative Hypothesis- Another theory Research hypothesis There is a difference Notation: H1 or Ha Parameter is really different from what is specified. We also have another idea the “Alternative hypothesis” This is usually our research hypothesis. It is the idea that something new is going on. For notation we often use H1 or Ha as notation for the alternative hypothesis. To set up the alternative in terms of parameters we typically have an idea that the parameter is in some way different from what we specified in the null hypothesis.We also have another idea the “Alternative hypothesis” This is usually our research hypothesis. It is the idea that something new is going on. For notation we often use H1 or Ha as notation for the alternative hypothesis. To set up the alternative in terms of parameters we typically have an idea that the parameter is in some way different from what we specified in the null hypothesis.

    46. 46 Statistical Significance Term often used in research literature. Show that a new treatment is significantly better. Significantly more than 20% Statistical significance=> unlikely to occur just by chance. Established by conducting a hypothesis test. Another term that we often use is “Statistical Significance” This term is often used in research literature and sometimes in the popular press. For instance we might see that a new treatment is significantly better than something else. We might also see that a subject might get significantly more than 20% of the cards correct. So what do we mean by statistically significant? We mean that the results would not happen just by random chance. Another term that we often use is “Statistical Significance” This term is often used in research literature and sometimes in the popular press. For instance we might see that a new treatment is significantly better than something else. We might also see that a subject might get significantly more than 20% of the cards correct. So what do we mean by statistically significant? We mean that the results would not happen just by random chance.

    47. Logical process To establish significance we will use a logical process that is what we typically call proof by contradiction. This process involves making an assumption. Then we evaluate the data. If the results are likely given the assumption then we can reject the assumption. If on the otherhand the results are unlikely given the assumption that may mean our assumption is wrong. To establish significance we will use a logical process that is what we typically call proof by contradiction. This process involves making an assumption. Then we evaluate the data. If the results are likely given the assumption then we can reject the assumption. If on the otherhand the results are unlikely given the assumption that may mean our assumption is wrong.

    48. 48 Hypothesis tests logic: Assume null hypothesis is true and establish sampling distribution How much variability could happen just by chance? Look at the sample data and determine if it is likely given the null hypothesis. If it is unlikely then our assumption of the null hypothesis may be wrong This same logical process will be used to deal with hypothesis tests that will establish statistical significance. We will assume our null hypothesis is true. That will allow us to establish a sampling distribution. We would then look at the sample data and determine if it is likely given the null hypothesis. If it is unlikely then our assumption of the null hypothesis may be wrong.This same logical process will be used to deal with hypothesis tests that will establish statistical significance. We will assume our null hypothesis is true. That will allow us to establish a sampling distribution. We would then look at the sample data and determine if it is likely given the null hypothesis. If it is unlikely then our assumption of the null hypothesis may be wrong.

    49. Psychic Experiment If we outline the psychic experiment and the logical process we use. We assume the skeptical idea that the person is just guessing. Then we look at the results of the experiment. If the subject gets around 1/5th correct then we would have no evidence that the subject is not just guessing. If the subject gets significantly more than 1/5th correct then we might reject the idea that the person is just guessing. If we outline the psychic experiment and the logical process we use. We assume the skeptical idea that the person is just guessing. Then we look at the results of the experiment. If the subject gets around 1/5th correct then we would have no evidence that the subject is not just guessing. If the subject gets significantly more than 1/5th correct then we might reject the idea that the person is just guessing.

    50. Psychic experiment One subject was presented with 100 cards. The subject got 30 of the 100 cards correct. Would this occur just by random chance? How likely is it for a subject to get 30 or more cards correct if he is just guessing? Lets look at the specifics. The subject was presented with 100 cards. The subject got 30 correct. 30 out of 100 is just 30% we might not think this is too many and not much more than 20%. But we do need to ask if it would happen just by random chance. How likely is it for a person to get 30 or more correct just by random chance. Lets look at the specifics. The subject was presented with 100 cards. The subject got 30 correct. 30 out of 100 is just 30% we might not think this is too many and not much more than 20%. But we do need to ask if it would happen just by random chance. How likely is it for a person to get 30 or more correct just by random chance.

    51. Psychic experiment One subject was presented with 100 cards. The subject got 30 of the 100 cards correct. Would this occur just by random chance? How likely is it for a subject to get 30 or more cards correct if he is just guessing? Lets look at the specifics. The subject was presented with 100 cards. The subject got 30 correct. 30 out of 100 is just 30% we might not think this is too many and not much more than 20%. But we do need to ask if it would happen just by random chance. How likely is it for a person to get 30 or more correct just by random chance. Lets look at the specifics. The subject was presented with 100 cards. The subject got 30 correct. 30 out of 100 is just 30% we might not think this is too many and not much more than 20%. But we do need to ask if it would happen just by random chance. How likely is it for a person to get 30 or more correct just by random chance.

    52. If the person is just guessing there is a 0.0062 chance of getting 30 or more correct. Assume just guessing We know that the proportion is normal centered around 0.2 and has standard deviation of 0.04. Calculating the standard score we get 2.5. Using table A we find that the corresponding probability is 0.0062. If the person is just guessing there is a 0.0062 chance of getting 30 or more correct. This is a pretty small chance. In other words if the person is just guessing it is unlikely that they would get this many correct. This may indicate that the person is not just guessing. In other words we might say that 30% is significantly more than what we would expect if the person is just guessing. We can say that this result is statistically significant, which would indicate a result this big would not happen just by random chance. We know that the proportion is normal centered around 0.2 and has standard deviation of 0.04. Calculating the standard score we get 2.5. Using table A we find that the corresponding probability is 0.0062. If the person is just guessing there is a 0.0062 chance of getting 30 or more correct. This is a pretty small chance. In other words if the person is just guessing it is unlikely that they would get this many correct. This may indicate that the person is not just guessing. In other words we might say that 30% is significantly more than what we would expect if the person is just guessing. We can say that this result is statistically significant, which would indicate a result this big would not happen just by random chance.

    53. Terminology Test Statistic- numeric measure of distance from sample value to what is expected under the null hypothesis. Typically standardize to make easier comparison. i.e. Z-score Calculated assuming null hypothesis is true. This brings up some more terminology. The test statistic is a numeric measure of how far the sample value is from what is expected under the null hypothesis. The test statistic is typically in the form of a standard score and is calculated assuming the null hypothesis is true. This brings up some more terminology. The test statistic is a numeric measure of how far the sample value is from what is expected under the null hypothesis. The test statistic is typically in the form of a standard score and is calculated assuming the null hypothesis is true.

    54. 54 Probability value (p-value) p-value- probability of obtaining a result more extreme than the sample value ASSUMING the null hypothesis is true. Measure of how likely it is to get this type of sample if H0 is true. Method for evaluating evidence. If p-value is small=> perhaps H0 is wrong Another term is the probability value or p-value. The p-value is the probability of obtaining a result more extreme than the sample value ASSUMING the null hypothesis is true. This gives us a numeric measure of how likely it is to get this type of statistic if the null hypothesis is true. If the p-value is small then perhaps the null hypothesis is incorrect. This does bring up the question of how small is small? Another term is the probability value or p-value. The p-value is the probability of obtaining a result more extreme than the sample value ASSUMING the null hypothesis is true. This gives us a numeric measure of how likely it is to get this type of statistic if the null hypothesis is true. If the p-value is small then perhaps the null hypothesis is incorrect. This does bring up the question of how small is small?

    56. 56 Significance Level How small is small? We need a cut off point for the p-value. Indication of small. Significance level is the cutoff point. Notation: a “alpha” Typically 10% or less Most common values 0.05 or 0.01 More skeptical=> use a smaller cutoff. How small is a small p-value? We need a cutoff point to decide when the p-value is small. This cutoff point is what we refer to as a significance level. We use the notation alpha to indicate the significance level. Typically we use a value of 10% or less for the significance level. Most commonly we also use values of 0.05 or 0.01. the more skeptical we are about something the smaller we set the value of alpha. The key is that if the probability-value is less than alpha we would say the result is statistically significant. In this class to simplify things we will use 0.05 by default. How small is a small p-value? We need a cutoff point to decide when the p-value is small. This cutoff point is what we refer to as a significance level. We use the notation alpha to indicate the significance level. Typically we use a value of 10% or less for the significance level. Most commonly we also use values of 0.05 or 0.01. the more skeptical we are about something the smaller we set the value of alpha. The key is that if the probability-value is less than alpha we would say the result is statistically significant. In this class to simplify things we will use 0.05 by default.

    57. Basic Steps of hypothesis test Specify assumptions Specify null and alternative hypotheses. Use sample data to calculate test statistic (assuming null hypothesis is true.) Calculate probability-value If probability-value is small then reject the null hypothesis. In the previous section we outlined some basic steps that we will use for every statistical hypothesis test we examine. We will begin by specifying any assumptions we might make about the situation. For instance we might need to assume that we have a random sample from the population of interest. The second step is to specify the null and alternative hypothesis. We then use our sample data to specify a test statistic. This is calculated assuming the null hypothesis is correct. We then use that test statistic to calculate the probability value. If the probability-value is small then we reject the null hypothesis. In the next section we will put these steps to use to set up a formal test for a proportion.In the previous section we outlined some basic steps that we will use for every statistical hypothesis test we examine. We will begin by specifying any assumptions we might make about the situation. For instance we might need to assume that we have a random sample from the population of interest. The second step is to specify the null and alternative hypothesis. We then use our sample data to specify a test statistic. This is calculated assuming the null hypothesis is correct. We then use that test statistic to calculate the probability value. If the probability-value is small then we reject the null hypothesis. In the next section we will put these steps to use to set up a formal test for a proportion.

    58. 58 Test for proportion Subjects are asked a yes/no question. Hypothesis is about a specific proportion Notation: p0 – a specific proportion of interest (like 0.20) “pi-naught” Lets take a look at a test about a proportion. In this case we are asking our subjects a yes no question. We will set up our null and alternative hypotheses in terms of a population proportion. We will specify a value of the population proportion. We will use the notation pi with a subscript zero or what we call “pi-naught” as a place holder in our formulas. In an actual problem we will have a specific value we are interested in. We will plug in that value everywhere we see pi-naught. Lets take a look at a test about a proportion. In this case we are asking our subjects a yes no question. We will set up our null and alternative hypotheses in terms of a population proportion. We will specify a value of the population proportion. We will use the notation pi with a subscript zero or what we call “pi-naught” as a place holder in our formulas. In an actual problem we will have a specific value we are interested in. We will plug in that value everywhere we see pi-naught.

    59. Test for proportion Assumption We have a random sample from the population of interest. Sample size is large enough that the distribution of the sample proportion is approximately normal. Usually 30 or more. Lets begin with the first step of our process. For this test we will need to specify some assumptions. We assume we have a random sample from the population of interest. We also assume that the sample size is large enough that the distribution of the sample proportion is approximately normal. usually 30 or more Lets begin with the first step of our process. For this test we will need to specify some assumptions. We assume we have a random sample from the population of interest. We also assume that the sample size is large enough that the distribution of the sample proportion is approximately normal. usually 30 or more

    60. 60 Hypotheses Null Hypothesis H0: ? = ? 0 Alternative hypothesis: H1: ? > ? 0 H1: ? < ? 0 H1: ? ? ? 0 For hypotheses we will have a null hypothesis that the population proportion pi is equal to pi-naught.For hypotheses we will have a null hypothesis that the population proportion pi is equal to pi-naught.

    61. 61 Hypotheses Null Hypothesis H0: ? = ? 0 Alternative hypothesis: H1: ? > ? 0 H1: ? < ? 0 H1: ? ? ? 0 Remember that pi-naught is a place holder for a specific value that we will plug in. The value of pi-naught is also plugged into the alternative hypothesis. Remember that pi-naught is a place holder for a specific value that we will plug in. The value of pi-naught is also plugged into the alternative hypothesis.

    62. 62 Hypotheses Null Hypothesis H0: ? = ? 0 Alternative hypothesis: H1: ? > ? 0 H1: ? < ? 0 H1: ? ? ? 0 For the alternative we have three choices and we will choose between these based on the specific situation. So our alternative may indicate that the proportion is larger than we thought or that it is less than we thought. We might also think it is just different from the specified proportion. For the alternative we have three choices and we will choose between these based on the specific situation. So our alternative may indicate that the proportion is larger than we thought or that it is less than we thought. We might also think it is just different from the specified proportion.

    63. 63 Test Statistic For a test statistic we are just calculating a standard score. Our standard score formula is of the form value minus mean divided by the standard deviation. The value of interest is the sample proportion that we see in our sample. You will also notice that this statistic includes the pi-naught, which is the specific amount we specified in the null hypothesis. For a test statistic we are just calculating a standard score. Our standard score formula is of the form value minus mean divided by the standard deviation. The value of interest is the sample proportion that we see in our sample. You will also notice that this statistic includes the pi-naught, which is the specific amount we specified in the null hypothesis.

    64. 64 Test Statistic So we will plug in the specific value that was in the null hypothesis. So we will plug in the specific value that was in the null hypothesis.

    65. 65 p-value Found from Table I Found in direction of alternative hypothesis Greater than or less than The next thing we need to find is the probability value. This value is found by looking up our Z value in table A. The probability is found in the direction of the alternative hypothesis. So in other words if you have a greater than hypothesis we look for the probability above the test statistic. If you have a less than hypothesis then we look for the probability below the test statistic. The next thing we need to find is the probability value. This value is found by looking up our Z value in table A. The probability is found in the direction of the alternative hypothesis. So in other words if you have a greater than hypothesis we look for the probability above the test statistic. If you have a less than hypothesis then we look for the probability below the test statistic.

    66. 66 Conclusion If p-value is less than ? reject H0. To make our conclusion we will compare the p-value to the significance level. A basic rule of thumb that we will use is that If the p-value is less than alpha then we will reject the null hypothesis. To make our conclusion we will compare the p-value to the significance level. A basic rule of thumb that we will use is that If the p-value is less than alpha then we will reject the null hypothesis.

    68. t-distribution Similar to normal distribution More values near ends In the previous section we examined the t-distribution. In this section we will examine how the t-distribution is used in a hypothesis test. In the previous section we examined the t-distribution. In this section we will examine how the t-distribution is used in a hypothesis test.

    69. The t- distribution For a random sample size n from a normal population, the distribution of follows a t-distribution with n-1 degrees of freedom. To construct this test we will need to make use of a fact. If we have a random sample from a normal distribution the standard score will have a t-distribution with n-1 degrees of freedom. One key point to realize is that the population from which we take our sample needs to have a normal distribution. The other point is that the degrees of freedom will be the sample size minus 1.To construct this test we will need to make use of a fact. If we have a random sample from a normal distribution the standard score will have a t-distribution with n-1 degrees of freedom. One key point to realize is that the population from which we take our sample needs to have a normal distribution. The other point is that the degrees of freedom will be the sample size minus 1.

    70. Use this to create a test If we are sampling from a normal population And sample size is small and unknown ? Use test statistic that has a t-distribution if H0 is true So we can create a test of hypothesis. If we are sampling from a population that follows a normal distribution and we don’t know the value of sigma we can use the t-distribution as the distribution from which we will find our p-values.So we can create a test of hypothesis. If we are sampling from a population that follows a normal distribution and we don’t know the value of sigma we can use the t-distribution as the distribution from which we will find our p-values.

    71. Test for mean (? unknown) Assumption We have a random sample from the population of interest The population is normally distributed. Lets look at the steps of the hypothesis test. We begin with the assumptions. As usual we should assume we have a random sample from the population of interest. We also need to assume that the population is normally distributed. This means we need to assume that if we looked at every value in the population and put them in a histogram the histogram would closely follow a normal distribution.Lets look at the steps of the hypothesis test. We begin with the assumptions. As usual we should assume we have a random sample from the population of interest. We also need to assume that the population is normally distributed. This means we need to assume that if we looked at every value in the population and put them in a histogram the histogram would closely follow a normal distribution.

    72. Test for mean (? unknown) Assumption We have a random sample from the population of interest The population is normally distributed. Generally robust to this assumption if sample is not too small. If sample is large enough the Central Limit Theorem applies. As with the confidence intervals this procedure typically is robust to the assumption of normality. In other words if the assumption does not exactly hold the procedure will still work as long as the sample is not extremely small. And as before if the sample is large enough the central limit theorem would apply and this test would work for any shape of population.As with the confidence intervals this procedure typically is robust to the assumption of normality. In other words if the assumption does not exactly hold the procedure will still work as long as the sample is not extremely small. And as before if the sample is large enough the central limit theorem would apply and this test would work for any shape of population.

    73. Hypotheses H0: ? = ?0 H1: ? > ?0 H1: ? < ?0 H1: ? ? ?0 For hypotheses, we will specify our hypotheses in terms of the mean mu. As with the proportions we will use a place holder for a specific value. The value of mu-naught will be some specific amount. For hypotheses, we will specify our hypotheses in terms of the mean mu. As with the proportions we will use a place holder for a specific value. The value of mu-naught will be some specific amount.

    74. Hypotheses H0: ? = ?0 H1: ? > ?0 H1: ? < ?0 H1: ? ? ?0 Also as we did with the proportions we will have a choice of three different alternative hypotheses. We will choose the appropriate alternative based on the situation of interest. Also as we did with the proportions we will have a choice of three different alternative hypotheses. We will choose the appropriate alternative based on the situation of interest.

    75. Test Statistic Recall that our test statistic often has the form statistic minus hypothesized value divided by standard error of the statistic. This form is how we set up a standard score and gives us an idea of how far off our statistic is from what we would expect if the null hypothesis is true.Recall that our test statistic often has the form statistic minus hypothesized value divided by standard error of the statistic. This form is how we set up a standard score and gives us an idea of how far off our statistic is from what we would expect if the null hypothesis is true.

    76. Test Statistic Our test statistic is basically the same as what we used previously. In this case we call it a t-statistic because we will use the t-distribution for comparison. Our test statistic is basically the same as what we used previously. In this case we call it a t-statistic because we will use the t-distribution for comparison.

    77. Test Statistic As before we take the sample mean and subtract the hypothesized value. We divide by the sample standard deviation over the square root of n.As before we take the sample mean and subtract the hypothesized value. We divide by the sample standard deviation over the square root of n.

    78. p-value t-table using n-1 df. t-table only has some specific values and exact p-value may not be possible. Found in direction of alternative hypothesis Our p-value is found using the t-table with n-1 degrees of freedom. As we saw in the previous section we often can not find the exact p-value for a test statistic, but instead can find only the range that contains the probability. For instance we might be able to determine that the p-value was less than 0.005 but bigger than 0.001. As before the p-value is found in the direction of the alternative hypothesis. Our p-value is found using the t-table with n-1 degrees of freedom. As we saw in the previous section we often can not find the exact p-value for a test statistic, but instead can find only the range that contains the probability. For instance we might be able to determine that the p-value was less than 0.005 but bigger than 0.001. As before the p-value is found in the direction of the alternative hypothesis.

    79. Comparison If p-value is less than ? reject H0. As always we make our conclusion using the rule of thumb that if the p-value is less than alpha we reject the null hypothesis.As always we make our conclusion using the rule of thumb that if the p-value is less than alpha we reject the null hypothesis.

    80. Example Does your car get the gas mileage listed on the window sticker? Recently the EPA revised how it rates gas mileage on cars. A television news producer decided to test the window sticker ratings of a particular model of car. Many people are concerned about the mileage a car gets. As you may or may not know recently the EPA changed the way it estimates the mileage to make it more realistic.. A television news producer decided to test the window sticker ratings of a particular model of car. Many people are concerned about the mileage a car gets. As you may or may not know recently the EPA changed the way it estimates the mileage to make it more realistic.. A television news producer decided to test the window sticker ratings of a particular model of car.

    81. Example They obtained 40 randomly selected cars of that model and tested their gas mileage over a week of highway driving. According to the EPA rating the car should get 32 mpg on the highway. In the sample of 40 cars the producer found the average highway mileage was 31.3mpg with a standard deviation of 4.7mpg. According to the EPA rating the car should get 32 mpg on the highway. In the sample of 40 cars the producer found the average highway mileage was 31.3mpg with a standard deviation of 4.7mpg. According to the EPA rating the car should get 32 mpg on the highway. In the sample of 40 cars the producer found the average highway mileage was 31.3mpg with a standard deviation of 4.7mpg.

    82. Example They obtained 40 randomly selected cars of that model and tested their gas mileage over a week of highway driving. According to the EPA rating the car should get 32 mpg on the highway. In the sample of 40 cars the producer found the average highway mileage was 31.3mpg with a standard deviation of 4.7mpg. According to the EPA rating the car should get 32 mpg on the highway. In the sample of 40 cars the producer found the average highway mileage was 31.3mpg with a standard deviation of 4.7mpg. So we see that the sample average is less than what is claimed. The question we need to ask is if that difference could have happened just by random chance or is it something that is statistically significant? We can decide by using the hypothesis test. According to the EPA rating the car should get 32 mpg on the highway. In the sample of 40 cars the producer found the average highway mileage was 31.3mpg with a standard deviation of 4.7mpg. So we see that the sample average is less than what is claimed. The question we need to ask is if that difference could have happened just by random chance or is it something that is statistically significant? We can decide by using the hypothesis test.

    83. Example Assumptions: We assume we have a random sample. We assume gas mileage is approximately normal This sample is reasonably large and this assumption is not a concern. We will begin with our assumptions. We assume we have a random sample. In this case the problem mentioned that the sample was random. We also assume gas mileage is approximately normal, although we may not know that this is true but since this sample is large the procedure should be robust to this assumption. We will begin with our assumptions. We assume we have a random sample. In this case the problem mentioned that the sample was random. We also assume gas mileage is approximately normal, although we may not know that this is true but since this sample is large the procedure should be robust to this assumption.

    84. Example Hypotheses: H0: µ = 32 (as the EPA specified) H1: µ < 32 (less than claimed) we need to specify the hypotheses. We will begin by assuming the status quo. The status quo is that the EPA is correct. This would mean that the average is 32 mpg as they claim. The alternative that the TV producer believes is that the average is something less than specified.we need to specify the hypotheses. We will begin by assuming the status quo. The status quo is that the EPA is correct. This would mean that the average is 32 mpg as they claim. The alternative that the TV producer believes is that the average is something less than specified.

    85. Example Test Statistic To calculate our test statistic we will plug in the appropriate values. Remember that the y-bar comes from the sample and mu-naught comes from the null hypothesis. Looking back to the null hypothesis we see that mu-naught is 32. Putting in the appropriate values we find that the test statistic is negative 0.94.To calculate our test statistic we will plug in the appropriate values. Remember that the y-bar comes from the sample and mu-naught comes from the null hypothesis. Looking back to the null hypothesis we see that mu-naught is 32. Putting in the appropriate values we find that the test statistic is negative 0.94.

    86. Example P-value DF=n-1=40-1=39 Not on table. Use 40 as closest row. Remember t-distribution is symmetric Probability below -0.94 is same as above 0.94 To find the probability value we go to table B. To do that we will need the appropriate degrees of freedom. Notice that we have an alternative hypothesis that is a less than alternative. That tells us we will be looking for the percentage below the negative 0.94.To find the probability value we go to table B. To do that we will need the appropriate degrees of freedom. Notice that we have an alternative hypothesis that is a less than alternative. That tells us we will be looking for the percentage below the negative 0.94.

    87. Table B We will need to go to our table and look only at the line with 40 degrees of freedom. Then find where our test statistic is located on this row. Again we will be looking only at the positive value remembering that this will be equivalent because of symmetry.We will need to go to our table and look only at the line with 40 degrees of freedom. Then find where our test statistic is located on this row. Again we will be looking only at the positive value remembering that this will be equivalent because of symmetry.

    88. Table B We see that the smallest value in the table is 1.303 so 0.94 would be to the left of this value.We see that the smallest value in the table is 1.303 so 0.94 would be to the left of this value.

    89. Table B That indicates that the probability would be to the left of the probability for the probability of 1.303, which is 0.100 or 10%. That indicates that our p-value would be greater than 0.100That indicates that the probability would be to the left of the probability for the probability of 1.303, which is 0.100 or 10%. That indicates that our p-value would be greater than 0.100

    90. Example P-value P(t<-0.94)= P(t>0.94)>0.100 Could find exact p-value using a computer. So we find that our probability value would be greater than 0.100. Note we can only find the approximate p-value using the table from our text. However, if we want a more exact p-value we can use a computer program such as SPSS, SAS or statcrunch to find the exact probability. In most real world situations we would use computer software to find those p-values.So we find that our probability value would be greater than 0.100. Note we can only find the approximate p-value using the table from our text. However, if we want a more exact p-value we can use a computer program such as SPSS, SAS or statcrunch to find the exact probability. In most real world situations we would use computer software to find those p-values.

    91. Example Conclusion: If p-value is less than ? reject H0. P-value>0.100 > 0.05 thus we do not Reject H0 No evidence that the average highway mileage is less than EPA claim Our conclusion will be based on the rule of thumb: if the p-value is less than alpha we reject the null hypothesis. In this case the p-value is greater than 0.100 is more than 0.05, so we will reject the null hypothesis. In other words an sample average of 31.3 is not unusual if the population average was 32. Our conclusion will be based on the rule of thumb: if the p-value is less than alpha we reject the null hypothesis. In this case the p-value is greater than 0.100 is more than 0.05, so we will reject the null hypothesis. In other words an sample average of 31.3 is not unusual if the population average was 32.

    92. P-values and alternatives: P-value is found in the direction of the alternative Greater than Less than What about not equal?

    93. Two sided alternatives H1: ? ? ? 0 H1: ? ? ?0 What do we do with these? p-value- probability of obtaining a result MORE EXTREME than the sample value assuming the null hypothesis is true Need to worry about more extreme in either direction

    94. Two sided alternatives Lets visualize the normal distribution and our test statistic. The distribution represents what would happen under the null hypothesis. If the null hypothesis is true we would expect values for the test statistic around 0. If we want our p-value we would look for what is away from 0. So we would look for what is outside of the test statistic. Lets visualize the normal distribution and our test statistic. The distribution represents what would happen under the null hypothesis. If the null hypothesis is true we would expect values for the test statistic around 0. If we want our p-value we would look for what is away from 0. So we would look for what is outside of the test statistic.

    95. Two sided alternatives So the p-value would be what is out here. So the p-value would be what is out here.

    96. Two sided alternatives But we are also worried about something that would be that extreme on the low side. So if we are looking at a test statistic we must find the equal probability from the other side of the distribution. But we are also worried about something that would be that extreme on the low side. So if we are looking at a test statistic we must find the equal probability from the other side of the distribution.

    97. Example We are testing the following hypotheses: H0: ? = 0.40 H1: ? ? 0.40 We find a test statistic of Z=2.73 Lets take a look at a simple example. We are going to test the null hypothesis that pi is 0.40 or 40% against the alternative that pi is not equal to 40%. In this case we have a test statistic that is 2.73. We need to find the appropriate probability value for this case.Lets take a look at a simple example. We are going to test the null hypothesis that pi is 0.40 or 40% against the alternative that pi is not equal to 40%. In this case we have a test statistic that is 2.73. We need to find the appropriate probability value for this case.

    98. Two sided alternatives Looking up the test statistic in the table we find 0.0032. That is the probability above our test statistic. Looking up the test statistic in the table we find 0.0032. That is the probability above our test statistic.

    99. Two sided alternatives But we also need what is below negative 2.73. That is an equal amount. So the total p-value is double the amount above the statistic.But we also need what is below negative 2.73. That is an equal amount. So the total p-value is double the amount above the statistic.

    100. Example P-value= P(Z>2.73)+P(Z<-2.73) = 2*P(Z>2.73)= 0.0064 So the total probability is the probability above the test statistic and the probability below the negative of that statistic. Since the normal distribution is symmetric this would just be twice the probability So the total probability is the probability above the test statistic and the probability below the negative of that statistic. Since the normal distribution is symmetric this would just be twice the probability

    101. Two sided alternatives If we have a two sided alternative we find the one sided p-value and double it. Since the normal distribution is symmetric so the probability above the test statistic would be the same as what is below. That means we simply need to find the p-value as usual and then double it. Since the normal distribution is symmetric so the probability above the test statistic would be the same as what is below. That means we simply need to find the p-value as usual and then double it.

    102. Class Problem: State whether each of the following assertions is a legitimate statistical hypothesis and why: H: s > 100 H: H: H: H: H: H: , where ? is the parameter of an exponential distribution used to model component lifetime.

    103. 103 Notes about hypothesis tests Statistical tests are based on randomness and assumptions. non-random experiments result in p-values that have questionable interpretation. Some other notes about hypothesis tests. One important point is that statistical tests are based on randomness and assumptions. If those assumptions are not valid the tests will not reflect the appropriate p-values. So if we have a non-random sample or a non-randomized experiment then the resulting p-values are not necessarily valid.Some other notes about hypothesis tests. One important point is that statistical tests are based on randomness and assumptions. If those assumptions are not valid the tests will not reflect the appropriate p-values. So if we have a non-random sample or a non-randomized experiment then the resulting p-values are not necessarily valid.

    104. 104 Notes about hypothesis tests Statistical Significance is not practical significance. Small p-value does not mean that it will really make a difference in a practical case. Significantly higher income (p-value = 0.0003) but it was $20.03 per year more. Statistical significance does not necessarily mean practical significance. A small p-value does not mean that it will really make a difference in a practical setting. For example we might find that the mean income is significantly higher for people who went to a particular school. But if that difference is about $20 per year then that amount may not really make a practical difference. So even though it has a very small p-value it does not mean that it is practically significant.Statistical significance does not necessarily mean practical significance. A small p-value does not mean that it will really make a difference in a practical setting. For example we might find that the mean income is significantly higher for people who went to a particular school. But if that difference is about $20 per year then that amount may not really make a practical difference. So even though it has a very small p-value it does not mean that it is practically significant.

    105. 105 Types of Errors What if we make the wrong conclusion? Type I error- reject the null hypothesis when it is true Some other items related to hypothesis testing. When we make a decision based on a hypothesis test we may not make the correct decision. This would mean we have made an error. We label the types of errors that may occur. A type I error occurs when we reject the null hypothesis when it is true. This can happen if we just happen by chance to get a test statistic that is out in the extreme parts of the distribution. So if we happen to get a value out near the end of the distributions even though the null hypothesis is true then we make a type I error.Some other items related to hypothesis testing. When we make a decision based on a hypothesis test we may not make the correct decision. This would mean we have made an error. We label the types of errors that may occur. A type I error occurs when we reject the null hypothesis when it is true. This can happen if we just happen by chance to get a test statistic that is out in the extreme parts of the distribution. So if we happen to get a value out near the end of the distributions even though the null hypothesis is true then we make a type I error.

    106. 106 Significance Level Significance level=> is the probability of a type I error. By keeping it small we reduce the chance of a type I error. Make it too small we may increase chance of a type II error. So how often will we make a type I error? The significance level “alpha” is the probability of the type I error. We keep the chance of the type I error small by keeping alpha small. But we need to be careful. If we make the chance of a type I error too small we increase the chance of a type II error.So how often will we make a type I error? The significance level “alpha” is the probability of the type I error. We keep the chance of the type I error small by keeping alpha small. But we need to be careful. If we make the chance of a type I error too small we increase the chance of a type II error.

    107. 107 Types of Errors Type II Error- fail to reject the null hypothesis when it is false. Miss a difference that is really there. A type II error occurs if we fail to reject the null when it is false. In other words we miss a difference that is really there.A type II error occurs if we fail to reject the null when it is false. In other words we miss a difference that is really there.

    108. 108 Power Power-how likely is the test to find a difference if one exists Chance of NOT making a type II error Another term we often associate with hypothesis testing is Power. The power of a test is the chance of not making a type II error. In other words Power is the chance of finding a difference if one exists. Another term we often associate with hypothesis testing is Power. The power of a test is the chance of not making a type II error. In other words Power is the chance of finding a difference if one exists.

    109. 109 Power Power-how likely is the test to find a difference if one exists Chance of NOT making a type II error The key is that Power is the chance of finding a difference if one really existsThe key is that Power is the chance of finding a difference if one really exists

    110. 110 Power Power-how likely is the test to find a difference if one exists Chance of NOT making a type II error Power=1-Probability(type II error) Power can also be thought of as 1 minus the probability of a type II error. Power can also be thought of as 1 minus the probability of a type II error.

    111. Increasing Power Power increases if we increase the sample size. Decreases the standard error of the statistic Power increases if we increase the significance level Use 0.10 instead of 0.05. Increases the chance of the type I error. In research studies we are often interested in increasing the power of a hypothesis test. The best way to increase power is to increase the sample size. A bigger sample will decrease the standard error of the statistic and make our hypothesis test more likely to detect a difference. One point to be aware of is that power can also be increased by increasing the value of alpha. The problem is that it will also increase the chance of a type I error. In research studies we are often interested in increasing the power of a hypothesis test. The best way to increase power is to increase the sample size. A bigger sample will decrease the standard error of the statistic and make our hypothesis test more likely to detect a difference. One point to be aware of is that power can also be increased by increasing the value of alpha. The problem is that it will also increase the chance of a type I error.

    112. Class Problem A new design for the breaking system on a certain type of car has been proposed. For the current system , the true breaking system, µ =120 feet at 40 mph. It is proposed that the new design be implemented only if sample data strongly indicates a reduction in true average breaking distance for the new design. State the relevant hypotheses, and describe the type 1 and type ii errors in the context of this situation.

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