Stats homework please help

Question

In a certain school district, it was observed that 32% of the students in the element schools were classified as only children (no siblings).  However, in the special program for talented and gifted children, 123 out of 326 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  level of significance.What is the hypothesized population proportion for this test? 

Answer 1

n = the total number of students in the elementary schools

p-hat = the sample proportion of students who are only children

p_o = the hypothesized proportion of students in the special program who are only children

H_o: p-hat = p_o

H_A: p-hat ≠ p_o

n = 326 elementary school students

p-hat = 32/100 = 0.32

po = 123/326 = 0.3773

Let's go on ahead and use the one-proportion z-test.

z = (p-hat) - p_o / √(p_o)(1 - p_o) / n = 0.32 - 0.3773 / √(0.3773)(0.6227)/326 = -0.0573006/0.0268457 = -2.13

P(z ≤ -2.13) = 0.0166 and P(z > 2.13) = 0.0166. Add them since we are doing the 2-sided test. The p-value will be 0.0332 > 0.02.

Since the p-value is greater than the significance level, we accept the null hypothesis. There is no sufficient evidence that shows the proportion of only children in the special program is significantly different from the proportion for the school district.

Answer 2

The null hypothesis is Ho: p = 0.32.

The alternate hypothesis is Ha: p /= 0.32

The sample proportion (phat) is 123/326 = 0.377.

The test statistic for the one-sample proportion is z = (phat - p)/sqrt(p*(1-p)/n), where n = 326.

z = (0.377 - 0.32)/(sqrt(0.32*0.68/326)) = 2.21

Since this is a 2-sided test, the p-value = P(|z| > 2.21) or P(z > 2.21) + P(z < -2.21). Since the normal distribution is symmetric, this is equal to 2 * P(z < -2.21) = 2 * 0.136 = 0.272.

Since the p-value = 0.272 > level of significance (0.02), we fail to reject Ho and conclude there is insufficient evidence that the proportion of only children in the special program is different from that of the school district.