Question

1. (harder) a group of 29 randomly selected students attend a special seminar on study skills. Of these 29, only 5 fail both. State a sensible null hypothesis, test it, and interpret.

Answer 1

Solution:

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

The sensible null and alternative hypotheses for this test can be taken as below:

Null hypothesis: H₀: The population proportion of students who fail both is 5%.

Alternative hypothesis: H_a: The population proportion of students who fail both is not 5%.

H₀: p = 0.05 versus H_a: p ≠ 0.05

This is a two tailed test.

We 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 = 5

n = sample size = 29

p̂ = x/n = 5/29 = 0.172413793

p = 0.05

q = 1 - p = 0.95

Z = (p̂ - p)/sqrt(pq/n)

Z = (0.172413793 - 0.05)/sqrt(0.05*0.95/29)

Z = 3.0247

Test statistic = 3.0247

P-value = 0.0025

(by using z-table)

P-value < α = 0.05

So, we reject the null hypothesis

There is not sufficient evidence to conclude that the population proportion of students who fail both is 5%.