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Statistics

Statistics. Hypothesis Tests. Contents. Developing Null and Alternative Hypotheses Type I and Type II Errors Population Mean: σ Known Population Mean: σ Unknown Population Proportion Hypothesis Testing and Decision Making

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Statistics

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  1. Statistics Hypothesis Tests

  2. Contents • Developing Null and Alternative Hypotheses • Type I and Type II Errors • Population Mean: σ Known • Population Mean: σUnknown • Population Proportion • Hypothesis Testing and Decision Making • Calculating the Probability of Type II Errors • Determining the Sample Size for • Hypothesis Tests About a Population Mean

  3. Contents • Population Proportion • Hypothesis Testing and Decision Making • Calculating the Probability of Type II Errors • Determining the Sample Size for • Hypothesis Tests About a Population Mean

  4. STATISTICSin PRACTICE • John Morrell & Company is considered the oldest continuously operating meat manufacturer in the United States. • Market research at Morrell provides management with up to- date information on the company’s various products and how the products compare with competing brands of similar products.

  5. STATISTICSin PRACTICE • One research question concerned whether Morrell’s Convenient Cuisine Beef Pot Roast was the preferred choice of more than 50% of the consumer population. The hypothesis test for the research question is Ho: p ≤ .50 vs. Ha: p > .50 • In this chapter we will discuss how to formulate hypotheses and how to conduct tests like the one used by Morrell.

  6. Developing Null and Alternative Hypotheses • Hypothesis testing can be used to determine • whether a statement about the value of a • population parameter should or should not be • rejected. • The null hypothesis, denoted by Ho , is a • tentative assumption about a population • parameter.

  7. Developing Null and Alternative Hypotheses • The alternative hypothesis, denoted by Ha, • is the opposite of what is stated in the null • hypothesis. • The alternative hypothesis is what the test is attempting to establish.

  8. Developing Null and Alternative Hypotheses • Testing Research Hypotheses • Testing the Validity of a Claim • Testing in Decision-Making Situations

  9. Developing Null and Alternative Hypotheses • Testing Research Hypotheses • The research hypothesis should be expressed • as the alternative hypothesis. • The conclusion that the research hypothesis • is true comes from sample data that contradict • the null hypothesis.

  10. Developing Null and Alternative Hypotheses • Testing Research Hypotheses Example: Consider a particular automobile model that currently attains an average fuel efficiency of 24 miles per gallon. A product research group developed a new fuel injection system specifically designed to increase the miles-per-gallon rating.

  11. Developing Null and Alternative Hypotheses The appropriate null and alternative hypotheses for the study are: Ho: μ ≤ 24 Ha: μ > 24

  12. Developing Null and Alternative Hypotheses • Testing Research Hypotheses • In research studies such as these, the null and • alternative hypotheses should be formulated • so that the rejection of H0 supports the research conclusion. The research hypothesis therefore should be expressed as the alternative hypothesis.

  13. Developing Null and Alternative Hypotheses • Testing the Validity of a Claim • Manufacturers’ claims are usually given the • benefit of the doubt and stated as the null • hypothesis. • The conclusion that the claim is false comes • from sample data that contradict the null • hypothesis.

  14. Developing Null and Alternative Hypotheses • Testing the Validity of a Claim Example: Consider the situation of a manufacturer of soft drinks who states that it fills two-liter containers of its products with an average of at least 67.6 fluid ounces.

  15. Developing Null and Alternative Hypotheses • Testing the Validity of a Claim A sample of two-liter containers will be selected, and the contents will be measured to test the manufacturer’s claim. The null and alternative hypotheses as follows. Ho: μ ≥ 67. 6 Ha: μ < 67. 6

  16. Developing Null and Alternative Hypotheses • Testing the Validity of a Claim • In any situation that involves testing the validity • of a claim, the null hypothesis is generally based • on the assumption that the claim is true. The • alternative hypothesis is then formulated so that • rejection of H0will provide statistical evidence • that the stated assumption is incorrect. • Action to correct the claim should be considered • whenever H0 is rejected.

  17. Developing Null and Alternative Hypotheses • Testing in Decision-Making Situations • A decision maker might have to choose • betweentwo courses of action, one • associated with the null hypothesis and • another associated with the alternative • hypothesis.

  18. Developing Null and Alternative Hypotheses • Testing in Decision-Making Situations Example: Accepting a shipment of goods from a supplier or returning the shipment of goods to the supplier. Assume that specifications for a particular part require a mean length of two inches per part. The null and alternative hypotheses would be formulated as follows. H0 : μ = 2 Ha: μ≠ 2

  19. Developing Null and Alternative Hypotheses • Testing in Decision-Making Situations • If the sample results indicate H0 cannot be rejected and the shipment will be accepted. • If the sample results indicate H0 should be rejected the parts do not meet specifications. The quality control inspector will have sufficient evidence to return the shipment to the supplier.

  20. Developing Null and Alternative Hypotheses • Testing in Decision-Making Situations • We see that for these types of situations, action is taken both when H0cannot be rejected and when H0 can be rejected.

  21. Summary of Forms for Null and Alternative Hypotheses about a Population Mean • The equality part of the hypotheses always • appears in the null hypothesis.(Follows Minitab) • In general, a hypothesis test about the value of • apopulation mean  must take one of the • followingthree forms (where 0is the • hypothesized value of the population mean).

  22. Summary of Forms for Null and Alternative Hypotheses about a Population Mean One-tailed (lower-tail) One-tailed (upper-tail) Two-tailed

  23. Null and Alternative Hypotheses • Example: Metro EMS A major west coast city provides one of the most comprehensive emergency medical services in the world. Operating in a multiple hospital system with approximately 20 mobile medical units, the service goal is to respond to medical emergencies with a mean time of 12 minutes or less.

  24. Null and Alternative Hypotheses • Example: Metro EMS The director of medical services wants to formulate a hypothesis test that could use a sample of emergency response times to determine whether or not the service goal of 12 minutes or less is being achieved.

  25. Null and Alternative Hypotheses The emergency service is meeting the response goal; no follow-up action is necessary. H0:  The emergency service is not meeting the response goal; appropriate follow-up action is necessary. Ha: where: μ = mean response time for the population of medical emergency requests

  26. Type I and Type II Errors • Because hypothesis tests are based on sample data, we must allow for the possibility of errors. • A Type I error is rejecting H0 when it is true. • Applications of hypothesis testing that only control the Type I error are often called significance tests.

  27. Type I and Type II Errors Type I error • The probability of making a Type I error when the null hypothesis is true as an equality is called the level of significance. • The Greek symbol α(alpha) is used to denote the level of significance, and common choices for α are .05 and .01.

  28. Type I and Type II Errors Type I error • If the cost of making a Type I error is high, small values of α are preferred. If the cost of making a Type I error is not too high, larger values of α are typically used.

  29. Type I and Type II Errors Type II error • A Type II error is accepting H0 when it is false. • It is difficult to control for the probability of • making a Type II error. Most applications of • hypothesis testing control for the probability • of making a Type I error. • Statisticians avoid the risk of making a Type II error by using “do not reject H0” and not “accept H0”.

  30. Type I and Type II Errors Population Condition H0 True (m< 12) H0 False (m > 12) Conclusion Not rejectH0 (Concludem< 12) Correct Decision Type II Error Type I Error Correct Decision RejectH0 (Conclude m > 12)

  31. Steps of Hypothesis Testing Step 1.Develop the null and alternative hypotheses. Step 2.Specify the level of significance α . Step 3.Collect the sample data and compute the test statistic. p-Value Approach Step 4.Use the value of the test statistic to compute the p-value. Step 5.Reject H0 if p-value <α.

  32. Steps of Hypothesis Testing Critical Value Approach Step 4. Use the level of significance to determine the critical value and the rejection rule. Step 5. Use the value of the test statistic and the rejection rule to determine whether to reject H0.

  33. p-Value Approach to One-Tailed Hypothesis Testing • The p-value is the probability, computed using the test statistic, that measures the support (or lack of support) provided by the sample for the null hypothesis. The p-value is also called the observed level of significance. • If the p-value is less than or equal to the level of significance α , the value of the test statistic is in the rejection region. • Reject H0 if the p-value <α .

  34. Critical Value Approach to One-Tailed Hypothesis Testing • The test statistic z has a standard normal probability distribution. • We can use the standard normal probability • distribution table to find the z-value with an • area of α in the lower (or upper) tail of the • distribution.

  35. Critical Value Approach to One-Tailed Hypothesis Testing • The value of the test statistic that established • theboundary of the rejection region is called • thecritical value for the test. • The rejection rule is: • Lower tail: Reject H0 if z< - z α • Upper tail: Reject H0 if z>zα 

  36. One-Tailed Hypothesis Testing • Population Mean: σ Known • One-tailed testsabout a population mean take one of the following two forms. Lower Tail Test Upper Tail Test

  37. Sampling distribution of Lower-Tailed Test About a Population Mean:sKnown p-Value <a, so reject H0. • p-Value Approach a = .10 p-value 72 z z = -1.46 -za = -1.28 0

  38. Sampling distribution of Lower-Tailed Test About a Population Mean : sKnown p-Value <a, so reject H0. • p-Value Approach a = .04 p-Value 11 z za = 1.75 z = 2.29 0

  39. Sampling distribution of Lower-Tailed Test About a Population Mean : sKnown • Critical Value Approach Reject H0 a 1 Do Not Reject H0 z -za = -1.28 0

  40. Sampling distribution of Upper-Tailed Test About a Population Mean : sKnown • Critical Value Approach Reject H0  Do Not Reject H0 z za = 1.645 0

  41. One-Tailed Hypothesis Testing Example: • The label on a large can of Hilltop Coffee states that the can contains 3 pounds of coffee. The Federal Trade Commission (FTC) knows that Hilltop’s production process cannot place exactly 3 pounds of coffee in each can, even if the mean filling weight for the population of all cans filled is 3 pounds per can.

  42. One-Tailed Hypothesis Testing Example: • However, as long as the population mean filling weight is at least 3 pounds per can, the rights of consumers will be protected.

  43. One-Tailed Hypothesis Testing Step 1. Develop the null and alternative hypotheses. denoting the population mean filling weight and the hypothesized value of the population mean is μ0 = 3. The null and alternative hypotheses are H0 : μ ≥ 3 Ha: μ < 3 Step 2. Specify the level of significance α. we set the level of significance for the hypothesis test at α= .01

  44. One-Tailed Hypothesis Testing Step 3. Collect the sample data and compute the test statistic. Test statistic: Sample data: σ = .18 and sample size n=36, = 2.92 pounds.

  45. One-Tailed Hypothesis Testing Step 3. Collect the sample data and compute the test statistic. We have z = -2.67

  46. One-Tailed Hypothesis Testing • p-Value Approach Step 4. Use the value of the test statistic to compute the p-value. Using the standard normal distribution table, the area between the mean and z = - 2.67 is .4962. The p-value is .5000 - .4962 = .0038, Step 5. p-value = .0038 < α = .01 Reject H0.

  47. One-Tailed Hypothesis Testing • p-value for The Hilltop Coffee Study when sample mean = 2.92 and z = -2 .67.

  48. One-Tailed Hypothesis Testing Critical Value Approach Step 4. Use the level of significance to determine the critical value and the rejection rule. The critical value is z = -2.33.

  49. One-Tailed Hypothesis Testing Step 5. Use the value of the test statistic and the rejection rule to determine whether to reject H0. We will reject H0 if z <- 2.33. Because z = - 2.67 < 2.33, we can reject H0 and conclude that Hilltop Coffee is under filling cans.

  50. One-Tailed Tests About a Population Mean : sKnown • Example: Metro EMS The response times for a random sample of 40 medical emergencies were tabulated. The sample mean is 13.25 minutes. The population standard deviation is believed to be 3.2 minutes.

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