Research Statistics Test 2
1. The honors history instructor and the honors statistics instructor had a friendly rivalry. They wanted to compare the mean performance of the honors history class to the mean performance of the honors statistics class (i.e., you will have to compute two z for a sample mean statistics). The mean score of the 27 students in the history class on the standardized History Test was 56. The mean score of the 18 students in the statistics class on the standardized Statistics Test was 75. The national mean (i.e., µ) and standard deviation (i.e., σ) for the History Test was 52 and 9, respectively. The national mean (i.e., µ) and standard deviation (i.e., σ) for the Statistics Test was 69 and 11, respectively. Which honors class did better on their respective standardized test?
1. History test: z = (56-52)/ (9/√27) = 2.309; Stats test: z = (75-69)/ (11/√18) = 2.314; the performance of the stats class was slightly better than the performance of the history class; 2.309 < 2.314
10. The hypothesis testing process involves two mutually exclusive hypotheses. What are they and what does it mean to say that they are mutually exclusive?
10. The null and research hypotheses are mutually exclusive in that they are the exact opposites of each other. If one is true then the other is false.
11. Explain why researchers cannot test research hypotheses directly. Why is it necessary for them to use a null hypothesis?
11. Research hypotheses are too vague to be tested directly. For example, research hypotheses say that a treatment works but they do not predict exactly how well it works. Therefore, there is no specific sample mean to expect or to test. In contrast, the null hypothesis predicts a specific mean that can be tested. If the sample mean is different from the expected mean the probability that the difference occurred due to sampling error can be computed. If the probability is low than researchers conclude that the null hypothesis is probably false.
12. What sample mean should you expect if the null hypothesis is true?
12. Expected mean if the null is true = 69
13. State the null hypothesis for this study. (You may use symbols and words).
13. Null:µHSC ≤ 69
14. State the research hypothesis for this study. (You may use symbols and words).
14. Research: µHSC > 69
15. What z-score should you expect if the null hypothesis is true?
15. Expected z score if the null is true = 0
16. What is the z-score that defines the critical region for this study?
16. 1.65 defines the start of the critical region for this one-tailed .05 test
17. What is the obtained z-score for this study?
17. z = (75-69) / (11/√18) = 2.314
18. Determine if you should reject the null or fail to reject the null.
18. reject
19. Compute the effect size for this study, interpret it as small, medium or large.
19. d = 6/11 = .54; this is a medium effect size
2. The History Test had a national mean of µ = 52 and a standard deviation of σ = 9. Use the z for a sample mean formula to determine the probability that a random sample of 27 students will have mean test score of 55 or greater.
2. z = (55-52)/ (9/√27) = 1.732 ⇒ the probability that a random sample of 27 students would have a mean history test score of 55 or higher is .0418.
20. Summarize the results using APA format (i.e., one or two sentences with statistical information).
20. The mean performance of the 18 students in the honors statistics class (M = 75) was significantly better than the national mean (µ = 69; σ = 11), z = 2.31, p < .05, d = .54.
22. If you change the alpha level from .01 to .05 does each of the following increase, decrease or stay the same? (a) Type I error, (b) Statistical Power, (c) Type II error (hint, when in doubt, draw it out! Sketch some hypotheticals to test your intuitions).
22. If the alpha level changes from .01 to .05: (a) Type I error increases, (b) Statistical power increases and (c) Type II error decreases.
23. If you change the sample size from n = 10 to n = 50 does each of the following increase, decrease or stay the same? (a) Type I error, (b) Statistical Power, (c) Type II error
23. If the sample size changes from 10 to 50: (a) Type I error does not change, (b) Statistical power increases and (c) Type II error decreases.
24. If you increase the size of the treatment effect does each of the following increase, decrease or stay the same? (a) Type I error, (b) Statistical Power, (c) Type II error
24. If the size of the treatment effect increases: (a) Type I error does not change, (b) Statistical power increases and (c) Type II error decreases.
25. What is the purpose of a significance test (also called hypothesis test)?
25. The purpose of a significance test is to determine if a researcher should reject or fail to reject the null; it is to determine if there is sufficient evidence to reject the null hypothesis.
26. What is the purpose of an effect size estimate?
26. The purpose of effect size estimates is to describe a treatment as only slightly effective, moderately effective or very effective.
27. If you reject the null you might have made what type of error? What are the two things that can cause this type of error?
27. If you reject the null you might have made a Type I error. Confounds and sampling error can cause Type I errors.
28. If you fail to reject the null you might have made what type of error? What are the four things that can cause this type of error? What can researchers do to address each of these problems if they want to design a better study in the future?
28. If you fail to reject the null you might have made a Type II error. Confounds, having too much measurement error (i.e., large σ), having too small of a sample size, and sampling error can cause Type II errors.
29. Define each of the following terms: standard deviation, standard error of the mean, Type I error, Type II error, Statistical Power, critical region, critical value, obtained value, p-value, null hypothesis, and research hypothesis.
29. Standard deviation: the average distance all scores vary from their mean; standard error of the mean: the average distance all possible sample means vary from their mean with is always the population mean; Type I error: rejecting a true null hypothesis; Type II error: failing to reject a false null hypothesis; statistical power: rejecting a false null hypothesis; critical region: the area of the of a curve that defines which values will result in rejecting the null hypothesis; critical value: the value that defines the start of the critical region; obtained value: the computed statistic (e.g., z score); p-value: the probability of getting the obtained value or a more extreme value due to sampling error if the null hypothesis is true; null hypothesis: the hypothesis that states the treatment will have no effect on the dv; research hypothesis: the hypothesis that states the treatment will have an effect on the dv.
3. The History Test had a national mean of µ = 52 and a standard deviation of σ = 9. Which is more likely to occur (a) drawing a random sample of 27 students from the national population of high school students that has a mean equal to or greater than 55 or (b) drawing a random sample of 100 students from the national population of high school students that has a mean equal to or greater than 55? Explain your answer.
3. The sample of 27 would be more likely to have a sample mean of 55 or higher because it is the smaller sample and would therefore have more sampling error. More sampling error means that the sample mean could vary more easily from the expected population value of 52.
30. Be able to recognize each of the following in a research scenario: Type I error, Type II error, Statistical Power, critical region, critical value, obtained value, p-value, null hypothesis, and research hypothesis.
30. Be able to recognize each of the above terms in a research scenario.
31. Be able to identify whether to use a z for a sample mean or a z for an individual score when presented with a research situation.
31. If a scenario is dealing with sample means use the z for a sample mean formula.
32. Know how to use a p-value to determine if you should reject or fail to reject a null hypothesis.
32. If the p-value is less than the alpha value you should reject the null.
33. Know how to use an obtained value to determine if you should reject or fail to reject a null hypothesis.
33. If the obtained value is greater than the critical value you should reject the null.
4. In the previous question, you should have concluded that (a) is more likely. If you didn't or if your explanation did not include a comparison of the sampling error in both situations it should have. Compute the amount of expected sampling error with samples of 27 and 100 then use these values to explain why (a) is the correct answer to the previous question. In your answer, be sure to explain what each of the sampling error values you compute mean.
4. Sampling error for sample size of 27 = 9/√27 = 1.73 which means that the average sampling error for all samples of 27 people is 1.73 away from the actual population mean; Sampling error for sample size of 100 = 9/√100 = .9 which means that the average sampling error for all samples of 100 people is .9 away from the actual population mean.
5. Verbally describe how a distribution of sample means could be constructed out of a population of scores.
5. A distribution of sample means is made by computing the mean of every random sample of n size that is possible and then plotting all of these means.
6. The central limit theorem describes three important properties of all distributions of sample means of any given sample size. What are the three properties it describes?
6. The central limit theorem describes the shape, center and spread of the distribution of sample means.
7. According to the central limit theorem what shape do distributions of sample means tend to take? How is this shape tendency impacted by sample size? When is the shape of a distribution of sample means impacted by the shape of the original population? When is it NOT impacted?
7. Distributions of sample means tend to be normal in shape. As n increases, the shape gets closer to a normal shape. If N = 30 or greater the shape is essentially normal. If N ≥ 30 the distribution of sample means is normal in shape even if the original population from which it was created was not normal. However, if the sample size is small AND the original population distribution is not normal in shape the distributions of sample means will NOT have a normal shape.
8. According to the central limit theorem what is the mean of all distributions of sample means going to equal?
8. µ
9. According to the central limit theorem what is the standard deviation of all distributions of sample means going to equal? How is this value impacted by sample size? This value has a special name, what is it? Explain why this value so important to researchers?
9. The standard deviation of the distribution of sample means will equal σ/√N, called the standard error of the mean. As N increases the standard error of the mean decreases. The standard error of the mean is valuable to research because it represents the typical amount of sampling error that can be expected given N and σ. Typical sampling error is the denominator to many statistics that are used to test null hypotheses.
21. If you want more practice with the hypothesis testing process (i.e., questions 12-20) answer these same questions for the class of history honors students. You can find the statistical information you need in question 1 above. There will be a couple of these scenarios on the test.
a. Expected sample mean = 52 b. The honors history class mean is not significantly better than the population mean of 52; Null:µHHC ≤ 52 c. The honors history class mean is significantly better than the population mean of 52; Null:µHHC > 52 d. Zero e. 1.65 f. z = (56-52)/(9/√27) = 4/1.732 = 2.31 g. Reject null h. d =(56 - 52) / 9 = .444; medium i. The mean performance of the 27 students in the honors history class (M = 56) was significantly better than the national mean (µ = 52; σ = 9), z = 2.31, p < .05, d = .44.
