Question

for the following information, determine whether a normal sampling distribution can be used, where p is the population proportion, α is the level of significance, \hat{p} is the sample proportion, and n is the sample size. if it can be used, test the claim.

claim: p ≥ 0.24; α = 0.08. sample statistics: \hat{p} = 0.20, n = 110

...

if a normal sampling distribution can be used, decide whether to reject or fail to reject the null hypothesis and interpret the decision. choose the correct answer below.

a. reject the null hypothesis. there is not enough evidence to reject the claim.
b. fail to reject the null hypothesis. there is enough evidence to reject the claim.
c. fail to reject the null hypothesis. there is not enough evidence to reject the claim.
d. reject the null hypothesis. there is enough evidence to reject the claim.
e. a normal sampling distribution cannot be used.

Explanation:

Step1: Check Normal Sampling Condition

To use a normal sampling distribution for a proportion, we need \( np \geq 5 \) and \( n(1 - p) \geq 5 \). Here, \( p = 0.24 \), \( n = 110 \).
Calculate \( np = 110\times0.24 = 26.4 \geq 5 \) and \( n(1 - p)=110\times(1 - 0.24)=110\times0.76 = 83.6 \geq 5 \). So normal sampling is valid.

Step2: State Hypotheses

Null hypothesis \( H_0: p \geq 0.24 \) (claim), Alternative \( H_a: p < 0.24 \) (left - tailed test).

Step3: Calculate Test Statistic

The formula for the z - statistic for a proportion is \( z=\frac{\hat{p}-p}{\sqrt{\frac{p(1 - p)}{n}}} \).
Substitute \( \hat{p}=0.20 \), \( p = 0.24 \), \( n = 110 \):
\( \sqrt{\frac{0.24\times(1 - 0.24)}{110}}=\sqrt{\frac{0.24\times0.76}{110}}=\sqrt{\frac{0.1824}{110}}\approx\sqrt{0.001658}\approx0.0407 \)
\( z=\frac{0.20 - 0.24}{0.0407}=\frac{- 0.04}{0.0407}\approx - 0.98 \)

Step4: Find Critical Value

For \( \alpha = 0.08 \) (left - tailed), the critical z - value \( z_{\alpha}=-1.405 \) (from z - table, since \( P(Z < z_{\alpha})=0.08 \)).

Step5: Decision Rule

Since the test statistic \( z=-0.98 > z_{\alpha}=-1.405 \), we fail to reject \( H_0 \). Failing to reject \( H_0 \) means there is not enough evidence to reject the claim (\( H_0 \) is the claim here).

Answer:

C. Fail to reject the null hypothesis. There is not enough evidence to reject the claim.