Question

 

Researchers are always looking for methods to help students improve their mathematics test scores. A company recently announced that they have found a way to help college students complete a certain mathematics test more quickly. The company said that using a specific meditation method for 20 minutes, before taking a mathematics test, would help students complete the test more quickly. Researchers have done studies, with thousands of general college students, to see how long it takes them (in minutes) to complete the mathematics test. The results follow a normal distribution with a mean of 70 minutes and standard deviation of 4 minutes. A mathematics researcher wants to use a statistical test to decide if the meditation method is believable. That is, will it actually help students complete the test more quickly? Researcher conduct a hypothesis with 100 college students who complete the test after meditating for 20 minutes to determine if students who meditate before the test are able to complete the mathematics test more quickly than normal.

4. Which of the following pairs represent appropriate hypotheses for this problem?

a. H0: µ = 70, Ha: µ > 70

b. H0: µ = 20, Ha: µ < 20

c. H0: µ = 20, Ha: µ > 20

d. H0: µ = 70, Ha: µ < 70

5. Using a representative sample, the researcher found that a sample of college students who meditated before the test had a mean time of x ̅=60.5 minutes and a P-value of less than 0.001. What could she (the researcher) conclude for this result at the 1% significance level?

a. The researcher has statistical evidence that, for college students, meditation improves the time it takes to complete the test.

b. The researcher has proven that, for college students, meditation substantially improves the time it takes to complete the test.

c. The researcher does not have enough evidence to say that, for college students, meditation substantially improves the time it takes to complete the test.

Answer 1

Let be the true average time taken by college students who complete the test after meditating for 20 minutes. The researcher want ts to test  if students who meditate before the test are able to complete the mathematics test more quickly than normal time which has a mean of 70 minutes. That is, the researcher wants to test if This would be the alternative hypothesis as it should contain one of the inequalities

4. Which of the following pairs represent appropriate hypotheses for this problem?

ans:
d. H0: µ = 70, Ha: µ < 70

5. Using a representative sample, the researcher found that a sample of college students who meditated before the test had a mean time of x ̅=60.5 minutes and a P-value of less than 0.001. What could she (the researcher) conclude for this result at the 1% significance level?

This is a left tailed test (The alternative hypothesis has "<"). However, since the p-value is given, our rule for testing the hypothesis is as given below, which would be the same for one-tail/two tail tests.

Rule: We will reject the null hypothesis if the p-value is less than the significance level.

Here, the p-value is <0.001 and it is less than 1% significance level (). Hence we reject the null hypothesis.

ans:

a. The researcher has statistical evidence that, for college students, meditation improves the time it takes to complete the test.

note: option b: We never can prove (which is a 100% certainty) a hypothesis, we can only gain statistical evidence supporting the claim.