In a certain school district, it was observed that $35 \%$ of the students in the element schools were classified as only children (no siblings). However, in the special program for talented and gifted children, 161 out of 405 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?
\[
p=
\]
.35
$0^{\circ}$
(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:
\begin{tabular}{|c|c|c|}
\hline (A) & (B) & (C) \\
\hline$H_{0}: p=0.35$ & $H_{0}: p=0.35$ & $H_{0}: p=0.35$ \\
$H_{a}: p< 0.35$ & $H_{a}: p \neq 0.35$ & $H_{a}: p> 0.35$ \\
\hline
\end{tabular}
(A) (B) (C) (D) (E) (F) $0^{\circ}$
Using the normal approximation for the binomial distribution (without the continuity correction), what is the test statistic ( $z$-score) 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 $=$
(Report answer as a decimal accurate to 4 decimal places.)
Answer
The P-value for this sample is approximately \(\boxed{0.0449}\).
Step 1 ：The hypothesized population proportion for this test is \(\boxed{0.35}\).
Step 2 ：This hypothesis test would be a \(\boxed{\text{two-tailed test}}\).
Step 3 ：The correct pair of hypotheses for this situation would be \(\boxed{H_{0}: p=0.35 \text{ and } H_{a}: p \neq 0.35}\).
Step 4 ：The test statistic (z-score) for this sample based on the sample proportion is approximately \(\boxed{2.005}\).
Step 5 ：The P-value for this sample is approximately \(\boxed{0.0449}\).
