Question

A multiple choice test has four possible responses. To test whether this question was answered correctly by more people than would be expected due to chance, a hypothesis test of 400 student answers is conducted.

1. What is the null and alternative hypothesis (in words and symbols).
2. If 218 of the 400 students got the correct response, what is the p-value? Explain your method and TI commands.
3. What decision is made if a significance level of 0.01 is used?
4. Based on this decision, what would you conclude about the test question?

Answer 1

Solution:

Given: A multiple choice test has four possible responses.

Since only one answer out of 4 choices is correct,

hence, p = probability of getting correct answer by guessing answer = 1/4 = 0.25

We have to test whether this question was answered correctly by more people than would be expected due to chance, that is we have to test if proportion of correct answers is more than 25% or 0.25.

Part a) What is the null and alternative hypothesis (in words and symbols).

Hypothesis in words:

H₀: The question was answered correctly by 25% of people which would be equal to proportion expected due to chance

Vs

H₁: The question was answered correctly by more people than would be expected due to chance ( that is more than 25% of people answered correctly)

Hypothesis in Symbols:

H₀: p = 0.25

Vs

H₁: p > 0.25

Part b) If 218 of the 400 students got the correct response, what is the p-value? Explain your method and TI commands.

x = 218

n = 400

Use following steps in TI 84 plus calculator:

Press STAT

Select TESTS

Under TESTS select 1-prop Z Test

Enter numbers:

Click on Calculate and Press Enter.

p-value = 1.470717 E ^-42

Since this is scientific number and it has E^-42, so we need to move 42 decimal places to left side.

Thus we get p-value approximately 0

Thus p-value = 0.0000

Part c) What decision is made if a significance level of 0.01 is used?

Decision Rule:
Reject H0, if p-value < 0.01 level of significance, otherwise we fail to reject H₀.

Since p-value = 0.0000 < 0.01 level of significance, we reject null hypothesis H₀.

Part d) Based on this decision, what would you conclude about the test question?

We have strong evidence against the null hypothesis and in favor of alternative hypothesis that the question was answered correctly by more people than would be expected due to chance.