Question

It is thought that 12% of all students taking a particular course received a grade of A. In a sample of 155 students, it is found that 21 made an A. can we conclude that the ratio of students with grade of A is higher than 12%? To do so

a) State the null and alternative hypotheses.

b) Compute the test statistic-value.

c) Find the critical-value.

d) Identify the decision rule and express your decision.

Answer 1

Here, we have to use one sample z test for the population proportion.

The null and alternative hypotheses for this test are given as below:

Null hypothesis: H0: the ratio of students with grade of A is 12%.

Alternative hypothesis: Ha: the ratio of students with grade of A is higher than 12%.

H0: p = 0.12 versus Ha: p > 0.12

This is an upper tailed test.

We are given

Assume Level of significance = α = 0.05

Test statistic formula for this test is given as below:

Z = (p̂ - p)/sqrt(pq/n)

Where, p̂ = Sample proportion, p is population proportion, q = 1 - p, and n is sample size

x = number of items of interest = 21

n = sample size = 155

p̂ = x/n = 21/155 = 0.135483871

p = 0.12

q = 1 - p = 0.88

Z = (p̂ - p)/sqrt(pq/n)

Z = (0.135483871 - 0.12)/sqrt(0.12*0.88/155)

Z = 0.5932

Test statistic = 0.5932

P-value = 0.2765

(by using z-table)

Critical value = 1.6449

(by using z-table)

Decision rule: Reject H0 if test statistic Z > 1.6449

Test statistic < Critical value

P-value > α = 0.05

So, we do not reject the null hypothesis

There is not sufficient evidence to conclude that the ratio of students with grade of A is higher than 12%.