Question

It is believed that nearsightedness affects about 8% of all children. In a random sample of 194 children, 21 are nearsighted. Conduct a hypothesis test for the following question: do these data provide evidence that the 8% value is inaccurate?

Answer 1

Answer:

These data do not provide evidence that the 8% value is inaccurate, this at the significance level of 5%.

Step-by-step explanation:

Let p be the true proportion of children nearsighted. We want to test the next hypothesis:

$H_{0}: p = 0.08$ vs $H_{a}: p \neq 0.08$ (two-tailed alternative)

We have a large sample size of n = 194 children. Therefore, the test statistic is given by

 

$Z = \frac{\hat{p}-0.08}{\sqrt{0.08(0.92)/194}}$ which is normally distributed.

The observed value is  

$z = \frac{21/194-0.08}{\sqrt{0.08(0.92)/194}} = 1.4502$.  

The rejection region using a significance level of 0.05 is given by RR = {z | z < -1.96 or z > 1.96}. Because the observed value 1.4502 does not fall inside the rejection region, we fail to reject the null hypothesis.