Statistics 280

Psy 280 Spring 2019

Homework 6

INSTRUCTIONS: For all significance tests, use alpha (α) = .05.

1. Mr. I will take your money from Empty Promises LLC has developed a new GRE test preparation course that promises to “increase your GRE Quantitative score by 10 points.” The course costs $1000, and will only require 2 afternoons, and can work around your busy schedule. It is known that the mean GRE Quantitative subtest score = 151.3 in the population, with a standard deviation of 8.7. A sample of 100 students takes Mr. Iwilltakeyourmoney’s course, and as a group, they earn a mean quantitative GRE subtest score = 152. Complete the following steps to conduct a statistical test that addresses whether taking the Empty Promises course has any effect on GRE Quantitative test scores.

a. What is the Independent Variable (IV) with its respective levels and what is the Dependent Variable (DV) in this study? 1/2 point

b. Step 1: Write the null and alternative hypotheses in symbols. Explain why hypotheses are expressed in terms of population parameters and not sample statistics? 1/2 point

c. Step 2: Assume that we draw all possible random samples of 100 people from the population of all adults. What would be the appropriate standard error (SE) for a distribution of samples of size 100? Give an interpretation in words of what this number means. 1 point

d. Step 2 (continued): Draw a sampling distribution using samples of 100 (n = 100). Label the x-axis in both raw scores and z-scores. 1/2 point

e. What proportion of sample means (or z’s for a set of sample means) would exceed a z of +/- 2.00, if HO is true? ½ point

f. Step 2 (continued): What is the critical value for the test statistic, assuming α = .05? ½ point

g. Step 3: Find the obtained (i.e., computed) test statistic for a sample (n=100) with an observed sample mean of 152 1 point

h. Step 4: Make a statistical decision about the null hypothesis. Will you reject or fail to reject the null based on your sample data? Why? 1/2 point

i. Step 4 (continued): What do your results mean in the context of this study, in words a layman would understand. ½ pt

j. Continuing with this example, (i.e. the same population mean, standard deviation, etc.), calculate the effect size of the Empty Promises Course on GRE quantitative subtest scores compared to the population average GRE quantitative score (using Cohen’s d). How large of an effect size is this, small, medium, or large? 1 point

k. The following table lists the possible decisions that can be made based on the research study above. Label which boxes would be correct decisions and which would be errors (Type I and Type II). 1 pt

Truth

H0 True

There is NO difference between Scores earned by students in the Empty Promises Course and those in the general population

H0 False

There is a difference between scores earned by students in the Empty Promises Course and those in the general population

Statistical

Decision

Reject H0

There is a difference between The scores earned by students in the Empty Promises Course and the general population

Fail to reject H0

There is NO difference between The scores earned by students in the Empty Promises Course and the general population

l. What would a Type I error mean in the context of this study? 1/2 point

m. What would a Type II error mean in the context of this study? 1/2 point

n. Given our statistical decision about the null hypothesis (part h), what is the only type of error we could make? Why? ½ pt

o. Please describe the four assumptions we make when we perform significance tests? 1 point

2. What is the effect of increasing n on the likelihood of rejecting the null hypothesis in the study described in #1 above, assuming everything else stays the same? Explain why increasing n has this effect. Given the value of d, would it be worth increasing the n, assuming the value of d stayed the same? 1.5 points

3. Dr. Thereisnofreelunch hypothesizes that students who spend at least 200 hours studying for the GRE (that’s 10 hours per week for 20 weeks, or 20 hours per week for 10 weeks, etc) will earn higher scores than the national average. She recruits a sample of 36 students, and provides a workspace and study materials. She tells the students to study for 200 hours prior to taking the GRE (she keeps track to make sure everyone in her sample studies 200 hours). It is known that the mean Verbal GRE subtest score is 150.8 in the population, with an SD = 8.5. Among the students in Dr. Therisnofreelunch’s study, the mean Verbal GRE subtest score was 158. Complete all 4 steps of hypothesis testing to decide if there is a significant difference in GRE Verbal subtest scores between students who study 200 hours and the general population.

a. Step 1: Write out the null and alternative hypotheses, using the appropriate symbols 1/2 pt.

b. Step 2: Assume that we draw all possible random samples of 36 individuals from the population of all college students. What would be the appropriate standard error (SE) for a distribution of samples of size 36? ½ pt

c. Step 2 (continued): What is the critical value for this test statistic, assuming α = .05? 1/2 point

d. Step 3: Calculate the test statistic (i.e. obtained z) 1 pt.

e. Step 4: Make a statistical decision about the null. Will you reject or fail to reject the null based on your sample data? Why? 1/2 pt.

f. Step 4 (continued): What do your results mean in the context of this study, in words a layman would understand?1 pt

g. The following table lists the possible decisions that can be made based on the research study above. Label which boxes would be correct decisions and which would be errors (Type I and Type II). 1 pt

Truth

H0 True

Studying 200 hours does NOT affect scores on the Verbal GRE Subtest

H0 False

Studying 200 influences scores on the Verbal GRE Subtest

Statistical

Decision

Reject H0

Studying 200 hours affects scores on the GRE Subtest

Fail to reject H0

Studying 200 hours does NOT affect scores on the GRE Verbal Subtest

h. What would a Type I error mean in the context of this study? 1/2 point

i. What would a Type II error mean in the context of this study? 1/2 point

j. Given our statistical decision about the null hypothesis (part f), what is the only type of error we could make? Why? ½ pt

k. What would happen in this study if we adopted an alpha of .01, rather than .05? Would we reject or fail to reject the null hypothesis? What would happen if we used an alpha of .001? Again, would we make a different decision about the null hypothesis? In general, what is the effect of using a smaller alpha level on our likelihood of rejecting a null hypothesis, assuming everything else stays the same? 1 point

l. Continuing with this example, (i.e. the same population mean, standard deviation, etc.), calculate the effect size in this study (using Cohen’s d). How large of an effect size is this, small, medium, or large? Does studying have a small, medium, or large effect on GRE scores? 1 point