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 \(=0.018\).