Find the alpha level and the critical region.

Question

I think I understand it however I just want to be certain.

GRE scores are on a 130–170

μ = 140

σ = 15

n =100, students who took a GRE course to improve their test scores.

after the course the average score for this sample is M = 155.

I am trying to determine if there is evidence that the GRE course effected the scores of the treated n.

I would greatly appreciate any insight!!!

Answer 1

Some details seem to be left out here, so I'm going to work with a few of my own assumptions:

You gave me σ instead of s for standard deviation, so I'll assume you gave me a population standard deviation for all GRE scores instead of a sample standard deviation. Since I have a population standard deviation and a sample size greater than 30, I'll use a z-test for a single sample. Also, I'm using this hypothesis test since we're interested in whether or not test scores increased with the GRE class: H₀: μ = 140 v. H₁: μ > 140. Here, μ represents the population average GRE score for all GRE scores. Also, I need a significance level α. Such a value wasn't provided, so I'll use the level of significance α = 0.05 here. The level of significance chosen depends on what's given to you or on how often you're willing to make Type I error (here, deciding that the GRE test class makes for a statistically significant score increase from 140, when in reality it doesn't). Now, I can conduct a hypothesis test. The test is based on this decision: Reject the null if the z test statistic is greater than the critical value z_crit = 1.645 (since this is the z-score with area 0.05 above it, to its right; invNorm(1-0.05, 0, 1) calculates this in a TI-83 or TI-84 calculator), else, fail to reject the null hypothesis.

If I use a TI-83 or TI-84 calculator, I'll go to STAT --> TESTS --> Z-Test, and input the following:

Inpt: Stats

μ₀:140

σ: 15

xbar: 155

n: 100

μ: > μ₀

Calculate.

I get this output:

μ > 140

z = 10

p = 7.770332E-24

xbar = 155

n=100

The z test statistic is 10, which exceeds a critical value of z_crit = 1.645 (the calculator does the critical value part behind the scenes). Also, even without knowing the z_crit value, the p-value is about 7.77 x 10^-24. This is rather small, certainly smaller than the level of significance α = 0.05. When the p-value is less than the significance level chosen earlier, we reject the null hypothesis since there is sufficient evidence in favor of the alternative hypothesis. Even if I had picked a significance level of α = 0.01 earlier, my p-value wouldn't change, and it would still be far below significance.

From this setup for a hypothesis test, I would believe that there is sufficient evidence in favor of the hypothesis that states that the population mean of GRE scores increased by a statistically significant amount. This is highly likely to come from the GRE class, but a hypothesis test doesn't determine causality: it just presents evidence in favor of the most plausible hypothesis.

Also, in terms of practical significance, think to yourself: is the general, average possibility for a 15 point increase in GRE scores a good reason for someone to be convinced to take the GRE course? Another thing, the best kind of test to use for this problem would be a two-sample z-test or two-sample t-test (based on what kind of information is provided), but the summary statistics for a control group whose members didn't take the GRE course weren't provided, hence the single sample construction. Just some food for thought. :)

I hope this helps!