Question

Every semester, I would like for more than 75% of my students to score a 70 or higher on the first test. This semester, out of the 72 students who took the first test, 59 got at least a C (scored higher a 70 or higher). Is there sufficient evidence to conclude, at the 10% significance level, that more than 75% of the students got at least a C on the first exam? Find the p-value.

Identify the null and alternative hypotheses, test statistic, critical value(s) and critical region or p-value, as indicated, and state the final conclusion that addresses the problem. Show all seven steps.

Answer 1

Solution:
A claim in this question is that more than 75% of the students got at least a C on the first exam
So we can write null and alternative hypothesis can be written as
Null hypothesis H0: p = 0.75
Alternative hypothesis Ha: p>0.75
No. of sample (n) = 72
No. of student got at least C (X) = 59
Sample proportion(p^) = X/N = 59/72
Here we will use the Z test as np = (72*0.75) =54 and n(1-p)=(72*(1-0.75)) = 18, as np and n(1-p) are more than 10 So we will use one proportion Z test
So Test statistic can be calculated as
Test stat = (p^ - p)/sqrt(p*(1-p)/n)) = ((59/72)-0.75)/sqrt(0.75*(1-0.75)/72)) = 1.36
at alpha = 0.1, and this is a right-tailed test so the critical value can be found from the Z table as follows:
Z-critical value = 1.28
and the critical region is if the test statistic value is greater than 1.28 than we will reject the null hypothesis else do not reject the null hypothesis.
So we can reject the null hypothesis as test statistic value is greater than 1.28
The P-value from Z table is 0.0869
at alpha = 0.1, we can see that the p-value is less than alpha value so we can reject the null hypothesis and, we have significant evidence to support the claim that more than 75% of the students got at least a C on the first exam.