In a certain school district, it was observed that $27 \%$ of the students in the element schools were classified as only children (no siblings). However, in the special program for talented and gifted children, 104 out of 316 students are only children. The school district administrators want to know if the proportion of only children in the special program is significantly different from the proportion for the school district. Test at the $\alpha=0.02$ level of significance.
What is the hypothesized population proportion for this test?
\[
p=
\]
(Report answer as a decimal accurate to 2 decimal places. Do not report using the percent symbol.)
Based on the statement of this problem, how many tails would this hypothesis test have?
one-tailed test
two-tailed test
Choose the correct pair of hypotheses for this situation:
(A) (B) (C) (D) (E) (F) $0^{6} \checkmark$
Using the normal approximation for the binomial distribution (without the continuity correction), was is the test statistic for this sample based on the sample proportion?
$z=$
(Report answer as a decimal accurate to 3 decimal places.)
You are now ready to calculate the $P$-value for this sample.
$P$-value $=$

Step 1 ：The hypothesized population proportion for this test is given in the problem statement as 27% of the students in the elementary schools were classified as only children. So, the hypothesized population proportion is \(p=0.27\).

Step 2 ：Since we are testing if the proportion of only children in the special program is significantly different from the proportion for the school district, this is a two-tailed test.

Step 3 ：The null hypothesis is that the proportion of only children in the special program is equal to the proportion for the school district (0.27), and the alternative hypothesis is that the proportion of only children in the special program is not equal to the proportion for the school district. So, the correct pair of hypotheses for this situation is: \(H_0: p = 0.27\) and \(H_1: p \neq 0.27\).

Step 4 ：The test statistic for this sample based on the sample proportion can be calculated using the formula for the z-score: \(z = \frac{{p_{hat} - p}}{{\sqrt{\frac{{p*(1-p)}}{n}}}}\), where \(p_{hat}\) is the sample proportion, \(p\) is the population proportion, and \(n\) is the sample size. Substituting the given values, we get \(z=2.367\).

Step 5 ：The P-value for this sample can be calculated using the z-score and the standard normal distribution. The calculated P-value is \(P\)-value \(=0.018\).

Step 6 ：\(\boxed{\text{Final Answer: }}\) The hypothesized population proportion for this test is \(p=0.27\). This is a two-tailed test. The correct pair of hypotheses for this situation is: \(H_0: p = 0.27\) and \(H_1: p \neq 0.27\). The test statistic for this sample based on the sample proportion is \(z=2.367\). The P-value for this sample is \(P\)-value \(=0.018\).