Question

A doctor believes that more than 80% of 2-year-old boys are taller than 33 inches. A random sample of 300 2-year-old boys revealed that 250 of them are taller than 33 inches. Test the doctor’s belief at the 5% level of significance.

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 = 0.80 versus Ha: p > 0.80

This is an upper tailed test.

We are given

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 = 250

n = sample size = 300

p̂ = x/n = 250/300 = 0.833333333

p = 0.8

q = 1 - p = 0.2

Z = (p̂ - p)/sqrt(pq/n)

Z = (0.833333333 - 0.8)/sqrt(0.8*0.2/300)

Z = 1.4434

Test statistic = 1.4434

P-value = 0.0745

(by using z-table)

P-value > α = 0.05

So, we do not reject the null hypothesis

There is not sufficient evidence to conclude that more than 80% of 2-year-old boys are taller than 33 inches.