Hypothesised population proportion, test statistic, p-value?

Question

In a certain school district, it was observed that 27% of the students in the elementary schools were classified as only children (no siblings).  However, in the special program for talented and gifted children, 107 out of 344 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?

Using the normal approximation for the binomial distribution (without the continuity correction), what is the test statistic for this sample based on the sample proportion?

 

You are now ready to calculate the P-value for this sample.

P-value = 

Answer 1

The population proportion (p0) is 0.27.

the sample proportion (phat) is 107/344 = 0.31

sample size (n) = 344

z = (phat - p0)/sqrt(p0 * (1-p0)/n( = (0.31 - 0.27)/sqrt(0.27*0.73/344) = 1.67

this is a two sided test so p-value = P(z > 1.67) + P(z < -1.67).

since the normal distribution is symmetric this is the same as 2*P(z < -1.67) = 2 * 0.0475 = 0.095.