Question

A hospital director is told that 55% of the treated patients are insured. The director wants to test the claim that the percentage of insured patients is below the expected percentage. A sample of 290 patients found that 145 were insured.

Make the decision to reject or fail to reject the null hypothesis at the 0.10 level.

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:

H0: p = 0.55 versus Ha: p < 0.55

This is a lower tailed test.

We are given

Level of significance = α = 0.10

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 = 145

n = sample size = 290

p̂ = x/n = 145/290 = 0.5

p = 0.55

q = 1 - p = 0.45

Z = (p̂ - p)/sqrt(pq/n)

Z = (0.50 - 0.55)/sqrt(0.55*0.45/290)

Z = -1.7115

Test statistic = -1.7115

P-value = 0.0435

(by using z-table)

P-value < α = 0.10

So, we reject the null hypothesis

There is sufficient evidence to conclude that the percentage of insured patients is below the expected percentage.