Question

A​ true/false test has 8080 questions. Suppose a passing grade is 5252 or more correct answers. Test the claim that a student knows more than half of the answers and is not just guessing. Assume the student gets 5252 answers correct out of 8080. Use a significance level of 0.05. Steps 1 and 2 of a hypothesis test procedure are given below. Show step​ 3, finding the test statistic and the​ p-value and step​ 4, interpreting the results.

Answer 1

Answer with explanation:

Let p be the proportion of the correct answers.

As per given , we have

$H_0: p=0.50$

$H_a: p>0.50$ , since the alternative hypothesis is right tailed , so the test is a one -tailed test.

If the student gets 52 answers correct out of 80.

i.e. the proportion of correct answers : $\dfrac{52}{80}=0.65$

Test statistic : $z=\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}$

$z=\dfrac{0.65-0.50}{\sqrt{\dfrac{0.50(0.50)}{80}}}\approx2.68$

P-value : $P(z>2.68)=0.0037$   [ by using p-value table for z (right-tailed)]

Since the p-value(0.0037) is less than the significance level (0.05), so we reject the null hypothesis.

Results : We have enough evidence to support the claim that a student knows more than half of the answers and is not just guessing.