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ITED 434 Quality Assurance

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ITED 434Quality Assurance

Statistics Overview: From HyperStat Online Textbook

http://davidmlane.com/hyperstat/index.html

by David Lane, Ph.D. Rice University

- Learn about the standard normal distribution
- Discuss descriptive and inferential statistics
- Learn how to calculate proportions under the normal curve.
- Discuss sampling distributions
- Learn how to calculate sample size from a normal distribution
- Discuss Hypothesis Testing 2 approaches:
- Classical method
- P-value method

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- The standard normal distribution is a normaldistributionwith a mean of 0 and a standard deviation of 1. Normal distributions can be transformed to standard normal distributions by the formula:
- X is a score from the original normal distribution, is the mean of the original normal distribution, and is the standard deviation of original normal distribution.

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- A z score always reflects the number of standard deviations above or below the mean a particular score is.
- For instance, if a person scored a 70 on a test with a mean of 50 and a standard deviation of 10, then they scored 2 standard deviations above the mean. Converting the test scores to z scores, an X of 70 would be:
- So, a z score of 2 means the original score was 2 standard deviations above the mean. Note that the z distribution will only be a normal distribution if the original distribution (X) is normal.

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Applying the formula will always produce a transformed variable with a mean of zero and a standard deviation of one. However, the shape of the distribution will not be affected by the transformation. If X is not normal then the transformed distribution will not be normal either. One important use of the standard normal distribution is for converting between scores from a normal distribution and percentile ranks.

Areas under portions of the standard normal distribution are shown to the right. About .68 (.34 + .34) of the distribution is between -1 and 1 while about .96 of the distribution is between -2 and 2.

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If a test is normally distributed with a mean of 60 and a standard deviation of 10, what proportion of the scores are above 85?

From the Z table, it is calculated that .9938 of the scores are less than or equal to a score 2.5 standard deviations above the mean. It follows that only 1-.9938 = .0062 of the scores are above a score 2.5 standard deviations above the mean. Therefore, only .0062 of the scores are above 85.

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The z table is used to determine that .9772 of the scores are below a score 2 standard deviations above the mean.

- Suppose you wanted to know the proportion of students receiving scores between 70 and 80. The approach is to figure out the proportion of students scoring below 80 and the proportion below 70.
- The difference between the two proportions is the proportion scoring between 70 and 80.
- First, the calculation of the proportion below 80. Since 80 is 20 points above the mean and the standard deviation is 10, 80 is 2 standard deviations above the mean.

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- The difference between the two proportions is the proportion scoring between 70 and 80.
- Next, calculate the proportion below 70. Note that the area of the curve below 70 is 1 standard deviation, or .1359
- To calculate the proportion between 70 and 80, subtract the proportion above 80 from the proportion below 70. That is .8413 - .0228 = .1359.
- Therefore, only 13.59% of the scores are between 70 and 80.

To calculate the proportion below 70:

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- Assume a test is normally distributed with a mean of 100 and a standard deviation of 15. What proportion of the scores would be between 85 and 105?
- The solution to this problem is similar to the solution to the last one. The first step is to calculate the proportion of scores below 85.
- Next, calculate the proportion of scores below 105. Finally, subtract the first result from the second to find the proportion scoring between 85 and 105.

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Begin by calculating the proportion below 85. 85 is one standard deviation below the mean:

Using the z-tablewith the value of -1 for z, the area below -1 (or 85 in terms of the raw scores) is .1587.

Do the same for 105

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The z-tableshows that the proportion scoring below .333 (105 in raw scores) is .6304. The difference is .6304 - .1587 = .4714. So .4714 of the scores are between 85 and 105.

Go to:http://davidmlane.com/hyperstat/z_table.htmlfor Z table.

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- If you compute the mean of a sample of 10 numbers, the value you obtain will not equal the population mean exactly; by chance it will be a little bit higher or a little bit lower.
- If you sampled sets of 10 numbers over and over again (computing the mean for each set), you would find that some sample means come much closer to the population mean than others. Some would be higher than the population mean and some would be lower.
- Imagine sampling 10 numbers and computing the mean over and over again, say about 1,000 times, and then constructing a relative frequency distribution of those 1,000 means.

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5 Samples

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10 Samples

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15 Samples

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20 Samples

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100 Samples

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1,000 Samples

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10,000 Samples

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- The distribution of means is a very good approximation to the sampling distribution of the mean.
- The sampling distribution of the mean is a theoretical distribution that is approached as the number of samples in the relative frequency distribution increases.
- With 1,000 samples, the relative frequency distribution is quite close; with 10,000 it is even closer.
- As the number of samples approaches infinity, the relative frequency distribution approaches the sampling distribution

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- The sampling distribution of the mean for a sample size of 10 was just an example; there is a different sampling distribution for other sample sizes.
- Also, keep in mind that the relative frequency distribution approaches a sampling distribution as the number of samples increases, not as the sample size increases since there is a different sampling distribution for each sample size.

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- A sampling distribution can also be defined as the relative frequency distribution that would be obtained if all possible samples of a particular sample size were taken.
- For example, the sampling distribution of the mean for a sample size of 10 would be constructed by computing the mean for each of the possible ways in which 10 scores could be sampled from the population and creating a relative frequency distribution of these means.
- Although these two definitions may seem different, they are actually the same: Both procedures produce exactly the same sampling distribution.

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- Statistics other than the mean have sampling distributions too. The sampling distribution of the median is the distribution that would result if the median instead of the mean were computed in each sample.
- Students often define "sampling distribution" as the sampling distribution of the mean. That is a serious mistake.
- Sampling distributions are very important since almost all inferential statistics are based on sampling distributions.

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- The sampling distribution of the mean is a very important distribution. In later chapters you will see that it is used to construct confidence intervals for the mean and for significance testing.
- Given a population with a mean of and a standard deviation of , the sampling distribution of the mean has a mean of and a standard deviation of s/ N , where N is the sample size.
- The standard deviation of the sampling distribution of the mean is called the standard error of the mean. It is designated by the symbol .

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- Note that the spread of the sampling distribution of the mean decreases as the sample size increases.

An example of the effect of sample size is shown above. Notice that the mean of the distribution is not affected by sample size.

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A variable's spread is the degree scores on the variable differ from each other.

If every score on the variable were about equal, the variable would have very little spread.

There are many measures of spread. The distributions on the right side of this page have the same mean but differ in spread: The distribution on the bottom is more spread out. Variability and dispersion are synonyms for spread.

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Notice that the graph is consistent with the formulas. If is sm= 10 for a sample size of 1 then sm should be equal to for a sample size of 25. When s is used as an estimate of σ, the estimated standard error of the mean is . The standard error of the mean is used in the computation of confidence intervals and significance tests for the mean.

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60

50

40

95 percent

upper confidence limit

30

20

10

0

60

80

90

100

10

20

30

40

50

70

N

-10

Number of tests

-20

-30

95 percent

lower confidence limit

-40

-50

Figure 11.3

Width of confidence interval versus number of tests.

-60

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SEE TABLE 11.1

Summary of confidence limit formulas

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SEE TABLE 10.6

Summary of common probability distributions.

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The central limit theorem states that given a distribution with a mean μ and variance σ2, the sampling distribution of the mean approaches a normal distribution with a mean (μ) and a variance σ2/N as N, the sample size, increases.

Go to Central Limit Demonstration:

http://oak.cats.ohiou.edu/~wallacd1/ssample.html

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- The central limit theorem also states that the larger our set of samples the more normal our distribution will be.
- Thus, the sampling distribution of the mean will have a normal shape and be come increasingly normal in shape as the number of samples increases.
- The sampling distribution of the mean will be normal regardless of the shape of the population distribution.
- Whether the population distribution is normal,positively or negatively skewed, unimodal or bimodal in shape,the sampling distribution of the mean will have a normal shape.

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- In the following example we start out with a uniform distribution. The sampling distribution of the mean, however, will contain variability in the mean values we obtain from sample to sample. Thus, the sampling distribution of the mean will have a normal shape, even though the population distribution does not. Notice that because we are taking a sample of values from all parts of the population, the mean of the samples will be close to the center of the population distribution.

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Hypothesis Testing

- The Classical Approach to hypothesis testing is to compare a test statistic and a critical value. It is best used for distributions which give areas and require you to look up the critical value (like the Student's t distribution) rather than distributions which have you look up a test statistic to find an area (like the normal distribution).
- The Classical Approach also has three different decision rules, depending on whether it is a left tail, right tail, or two tail test.
- One problem with the Classical Approach is that if a different level of significance is desired, a different critical value must be read from the table.

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- A null hypothesis is not accepted just because it is not rejected.
- Data not sufficient to show convincingly that a difference between means is not zero do not prove that the difference is zero.
- No experiment can distinguish between the case of no difference between means and an extremely small difference between means.
- If data are consistent with the null hypothesis, they are also consistent with other similar hypotheses.

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Left Tailed Test

H1: parameter < valueNotice the inequality points to the left Decision Rule: Reject H0 if t.s. < c.v.

Right Tailed Test

H1: parameter > valueNotice the inequality points to the right Decision Rule: Reject H0 if t.s. > c.v.

Two Tailed Test

H1: parameter not equal valueAnother way to write not equal is < or >Notice the inequality points to both sides Decision Rule: Reject H0 if t.s. < c.v. (left) or t.s. > c.v. (right)

The decision rule can be summarized as follows:

Reject H0 if the test statistic falls in the critical region

(Reject H0 if the test statistic is more extreme than the critical value)

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- The P-Value Approach, short for Probability Value, approaches hypothesis testing from a different manner. Instead of comparing z-scores or t-scores as in the classical approach, you're comparing probabilities, or areas.
- The level of significance (alpha) is the area in the critical region. That is, the area in the tails to the right or left of the critical values.
- The p-value is the area to the right or left of the test statistic. If it is a two tail test, then look up the probability in one tail and double it.
- If the test statistic is in the critical region, then the p-value will be less than the level of significance. It does not matter whether it is a left tail, right tail, or two tail test. This rule always holds.
- Reject the null hypothesis if the p-value is less than the level of significance.

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- You will fail to reject the null hypothesis if the p-value is greater than or equal to the level of significance.
- The p-value approach is best suited for the normal distribution when doing calculations by hand. However, many statistical packages will give the p-value but not the critical value. This is because it is easier for a computer or calculator to find the probability than it is to find the critical value.
- Another benefit of the p-value is that the statistician immediately knows at what level the testing becomes significant. That is, a p-value of 0.06 would be rejected at an 0.10 level of significance, but it would fail to reject at an 0.05 level of significance. Warning: Do not decide on the level of significance after calculating the test statistic and finding the p-value.

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- Any proportion equivalent to the following statement is correct:
The test statistic is to the p-value as the critical value is to the level of significance and the test is know as a “significance test.”

- The null hypothesis is rejected if p is at or below the significance level; it is not rejected if p is above the significance level.
- The degree to which p ends up being above or below the significance level does not matter.

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- Hypothesis testing is a method of inferential statistics.
- Researchers very frequently put forward a null hypothesis in the hope that they can discredit it.
- Data are then collected and the viability of the null hypothesis is determined in light of the data.
- If the data are very different from what would be expected under the assumption that the null hypothesis is true, then the null hypothesis is rejected.
- If the data are not greatly at variance with what would be expected under the assumption that the null hypothesis is true, then the null hypothesis is not rejected.

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- Note: Failure to reject the null hypothesis is not the same thing as accepting the null hypothesis.

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- 1. The first step in hypothesis testing is to specify the null hypothesis (H0) and the alternative hypothesis (H1). If the research concerns whether one method of presenting pictorial stimuli leads to better recognition than another, the null hypothesis would most likely be that there is no difference between methods (H0: µ1 - µ2 = 0). The alternative hypothesis would be H1: µ1= µ2. If the research concerned the correlation between grades and SAT scores, the null hypothesis would most likely be that there is no correlation (H0: ρ= 0). The alternative hypothesis would be H1: ρ0.
- 2. The next step is to select a significance level. Typically the .05 or the .01 level is used.
- 3. The third step is to calculate a statistic analogous to the parameter specified by the null hypothesis. If the null hypothesis were defined by the parameter µ1- µ2, then the statistic M1 - M2 would be computed.

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- 4. The fourth step is to calculate the probability value (often called the p value) which is the probability of obtaining a statistic as different or more different from the parameter specified in the null hypothesis as the statistic computed from the data. The calculations are made assuming that the null hypothesis is true. (click here for a concrete example)
- 5. The probability value computed in Step 4 is compared with the significance level chosen in Step 2. If the probability is less than or equal to the significance level, then the null hypothesis is rejected; if the probability is greater than the significance level then the null hypothesis is not rejected. When the null hypothesis is rejected, the outcome is said to be "statistically significant"; when the null hypothesis is not rejected then the outcome is said be "not statistically significant."

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- 6. If the outcome is statistically significant, then the null hypothesis is rejected in favor of the alternative hypothesis. If the rejected null hypothesis were that µ1- µ2 = 0, then the alternative hypothesis would be that µ1=µ2. If M1 were greater than M2 then the researcher would naturally conclude that µ1 µ2. (Click here to see why you can conclude more than µ1=µ2).
- 7. The final step is to describe the result and the statistical conclusion in an understandable way. Be sure to present the descriptive statistics as well as whether the effect was significant or not. For example, a significant difference between a group that received a drug and a control group might be described as follow:
- Subjects in the drug group scored significantly higher (M = 23) than did subjects in the control group (M = 17), t(18) = 2.4, p = 0.027.

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- The statement that "t(18) =2.4" has to do with how the probability value (p) was calculated. A small minority of researchers might object to two aspects of this wording.
- First, some believe that the significance level rather than the probability level should be reported. The argument for reporting the probability value is presented in another section.
- Second, since the alternative hypothesis was stated as µ1=µ2, some might argue that it can only be concluded that the population means differ and not that the population mean for the drug group is higher than the population mean for the control group.

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- This argument is misguided. Intuitively, there are strong reasons for inferring that the direction of the difference in the population is the same as the difference in the sample. There is also a more formal argument. A non-significant effect might be described as follows:
- Although subjects in the drug group scored higher (M = 23) than did subjects in the control group, (M = 20), the difference between means was not significant, t(18) = 1.4, p = .179.

- It would not have been correct to say that there was no difference between the performance of the two groups. There was a difference. It is just that the difference was not large enough to rule out chance as an explanation of the difference. It would also have been incorrect to imply that there is no difference in the population. Be sure not to accept the null hypothesis.

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