Question

We are testing people to see if the rate of use of seat belts has changed from a previous value of 82%. Suppose that in our random sample of 75 people, we see that 66 have the seat belt fastened. The researcher is interested in α = 0.1 level test.

Step 1 State the null and alternative hypotheses.

Step 2 Write down the appropriate test statistic and the rejection region of your test (report z critical(s))

Step 3 Compute the value of the test statistic (z observed)

Step 4: State your conclusion (in one sentence, state whether or not the test rejects the null hypothesis and in another sentence apply the results to the problem).

Step 5: Compute the p- value for this test. State your decision.

Answer 1

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

Step 1 State the null and alternative hypotheses.

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

Null hypothesis: H₀: the rate of use of seat belts is 82%.

Alternative hypothesis: H_a: the rate of use of seat belts is not 82%.

H₀: p = 0.82 versus H_a: p ≠ 0.82

This is a two tailed test.

Step 2

We are given

Level of significance = α = 0.1

Critical value = - 1.6449 and 1.6449

(by using z-table)

Reject H₀ if z < -1.6449 or z > 1.6449

Step 3

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 = 66

n = sample size = 75

p̂ = x/n = 66/75 = 0.88

p = 0.82

q = 1 - p = 0.18

Z = (p̂ - p)/sqrt(pq/n)

Z = (0.88 - 0.82)/sqrt(0.82*0.18/75)

Z = 1.3525

Test statistic = 1.3525

Step 4:

Test statistic value lies between critical values, so we do not reject H₀.

There is not sufficient evidence to conclude that the rate of use of seat belts has changed from a previous value of 82%.

Step 5:

P-value = 0.1762

(by using z-table)

P-value > α = 0.10

So, we do not reject the null hypothesis

There is not sufficient evidence to conclude that the rate of use of seat belts has changed from a previous value of 82%.