PDF 1. Define each of the following - Bryan Burnham

[Pages:4]1. Define each of the following:

a. null hypothesis b. alternate hypothesis c. critical value d. rejection region e. test statistic f. alpha level

2. A high school administration is interested in whether their students save more or less than $100 per month toward college. Translate this question into a null hypothesis and an alternate hypothesis.

3. Why can we never accept the null hypothesis as being true based on statistical tests?

4. Use the following information to answer the questions that follow: H0: = 5, H1: 5, X = 8 , = 15, and n =

25.

a. Using an alpha level of = .05, what is the critical value? b. Calculate the standard error of the mean. c. Calculate the test-statistic for the sample mean d. What decisions should you make regarding the null and alternate hypotheses?

5. Use the following information to answer the questions that follow: H0: = 6, H1: 6, X = 4.50 , = 3, and n =

36.

a. Using an alpha level of = .05, what is the critical value? b. Calculate the standard error of the mean. c. Calculate the test-statistic for the sample mean d. What decisions should you make regarding the null and alternate hypotheses?

6. Use the following information to answer the questions that follow: H0: = 6, H1: < 6, X = 4.50 , = 5, and n =

49.

a. Using an alpha level of = .05, what is the critical value? b. Calculate the standard error of the mean. c. Calculate the test-statistic for the sample mean d. What decisions should you make regarding the null and alternate hypotheses?

7. Use the following to answer the questions below: Scores on IQ tests are normally distributed with = 100 and = 15. I am interested in whether University of Scranton Psychology majors have a mean IQ greater than the general population. I randomly select n = 25 psychology majors and find they have a sample mean IQ of 104.

a. Expressed in terms of , what are the null and alternate hypotheses? b. Using an alpha level of = .05, what is the critical value? c. Calculate the standard error of the mean. d. What is the obtained z-Score for the sample mean of 104? e. What decisions should you make regarding the hypotheses? What should you conclude about the mean

IQ of University of Scranton psychology majors compared to the general population?

8. Use the following to answer the questions below: You are a faculty member at East Buzzkill University and you believe students at your institution poorer reading comprehension skills than other college students. You test you hypothesis by comparing SAT-Verbal scores of students attending East Buzzkill University to the mean SAT-Verbal score of all students taking the SATs. Scores on the SAT-Verbal test are normally distributed with = 500 and = 75. You sample n = 100 East Buzzkill University students and ask for their SAT-Verbal scores. The sample has a mean of mean of 480.

a. Expressed in terms of , what are the null and alternate hypotheses? b. Using an alpha level of = .05, what is the critical value? c. Calculate the standard error of the mean. d. What is the obtained z-Score for the sample mean of 480? e. Using an alpha level of .05, what decisions should you make regarding the hypotheses? What should you

conclude about the SAT verbal scores and reading comprehension abilities of East Buzzkill University students?

9. Define each of the following.

a. Type I error b. Type II error c. alpha-level d. beta e. power

10. What is the relationship between alpha and the probability of a Type I error. What is the reason for this relationship?

11. What is the relationship between power and the probability of a Type II error? What is the reason for this relationship?

12. What is the relationship between alpha and power? What is the reason for this relationship?

13. What effects does sample size have on the power of a statistical test?

14. Under what circumstance should a directional rather than a non-directional test be used? Why? Under what circumstance should a non-directional rather than a directional test be used? Why?

15. What effect does sample size have on the power of a statistical test? Why?

ANSWERS

1. .a The hypothesis that we assume is true for the purpose of conducting a hypothesis test. This predicts there will be no difference between a sample mean and a population mean. b. The hypothesis that is competing with the null hypothesis. This predicts that there will be a difference between a sample mean and the population mean. c. The endpoints (in z scores) of a range in which the sample mean is expected to fall if the null hypothesis is correct. It is the minimum value needed for a test-statistic to be considered statistically significant. d. The area of the normal distribution beyond the critical value; that is, the area that is more extreme than the critical values. If the sample mean falls in this region, it is an unexpected result. e. This is the z-score for the sample mean. f. The probability that is associated with being in the rejection region.

2.

H0: = $100

H1: $100

or

H0: HS = $100

or

H1: HS $100

3. Due to sampling error, we can never be completely certain that the true population mean is equal to any one specific value based on sample data.

4. a. +/-1.96 b. 3 c. 1 d. Retain null hypothesis and make no decision on alternate hypothesis.

5. a.+/- 1.96 b. 0.5 c. -3 d. Reject the null hypothesis and accept the alternate hypothesis.

6. a. -1.65

b. 0.714 c. -2.101 d. Reject the null hypothesis and accept the alternate hypothesis.

7. a. H0: Psychology = 100 H1: Psychology > 100

b. +1.65

c. 3

d. 1.333

e. The difference is not statistically significant and we retain null hypothesis. The results suggest the mean IQ for

University of Scranton psychology majors is not significantly greater than the general population.

8. a. H0: Buzzkill = 500

H1: Buzzkill < 500

b. -1.65

c. 7.5

d. -2.667

e. The difference is statistically significant and we reject the null hypothesis and accept the alternate hypothesis.

The results suggest students at East Buzzkill University have significantly lower SAT-Verbal scores than the

population of students taking the SAT-Verbal test, which suggests East Buzzkill University students have poor

reading comprehension skills.

9. a. When a true null hypothesis is incorrectly rejected b. When a false null hypothesis is incorrectly retained. c. The probability of making a Type I error. d. The probability of making a Type II error. e. Equal to 1 - , this is the probability of correctly rejecting a false null hypothesis.

10. The probability of committing a Type 1 error is equal to the chosen alpha level. This is because if the null hypothesis is true, it will be incorrectly rejected any time any time the test statistic falls inside the rejection region by chance. The probability of the test statistic falling in the rejection region is equal to the alpha level.

11. Power is equal to 1 minus the probability of making a Type II error, which is known as beta (). This is because the probability of a correct decision when the null hypothesis is false is equal to 1 - the probability of making a Type II error ().

12. The power of an inferential test decreases as the alpha level gets smaller. This is because as alpha gets smaller, the probability of rejecting the null hypothesis decreases.

13. Generally, holding all else constant, larger sample sizes are more statistically powerful compared to smaller sample sizes. This is so, because, as sample size increases, the size of the standard error decreases.

14. A directional test should be used when there is some reason to believe that the actual population mean and hypothesized population mean differ in a specific direction. A non-directional test should be used if there is no concern over the specific direction of difference between the hypothesized population mean and the true population mean.

15. The larger the sample size, the greater the power. This is because as the sample size increases, the size of the standard error term will decrease. As long as the mean difference stays the same, this will increase the size of the test-statistic.

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