Question

Given that the standard error of the sample proportion is 0.0195 and the point estimate follows a nearly normal distribution, calculate the test statistic (the z-statistic).

Answer 1

Given the question:

Nearsighted. It is believed that nearsightedness affects about 8% of all children. In a random sample of 194 children, 21 are nearsighted.

(a) Construct hypotheses appropriate for the following question: do these data provide evidence that the 8% value is inaccurate?

(b) What proportion of children in this sample are nearsighted?

(c) Given that the standard error of the sample proportion is 0.0195 and the point estimate follows a nearly normal distribution, calculate the test statistic (the Z-statistic).

(d) What is the p-value for this hypothesis test?

(e) What is the conclusion of the hypothesis test?

Part A:

The appropriate hypotheses for the question: do these data provide evidence that the 8% value is inaccurate is given by

$H_o:p=0.08 \\ \\ H_a:p\neq0.08$



Part B:

The proportion of children in this sample that are nearsighted is given by

$\frac{21}{194} =0.1082=10.82\%$



Part C

Given that the standard error of the sample proportion is 0.0195 and the point estimate follows a nearly normal distribution,

The test statistic is calculated as follows:

$\frac{\hat{p}-p}{SE_p} = \frac{0.1082-0.08}{0.0195} = \frac{0.0282}{0.0195} =1.446$

Therefore, the test statistic is 1.446.



Part D

The p-value for this hypothesis test is given by

$P(-1.446 \leq z \leq 1.446)=2P(z\leq-1.446)=2(0.0741)=0.1482$



Part E

Since the P-value (0.1482) is relatively large, we cannot reject the null hypothesis.

Therefore, we conclude that these data does not provide evidence that the 8% value is inaccurate.