Question

Several years​ ago, 39​% of parents who had children in grades​ K-12 were satisfied with the quality of education the students receive. A recent poll asked 1 055 parents who have children in grades​ K-12 if they were satisfied with the quality of education the students receive. Of the 1 055 ​surveyed, 466 indicated that they were satisfied. Construct a 95​% confidence interval to assess whether this represents evidence that​ parents' attitudes toward the quality of education have changed.

Answer 1

Answer with explanation:

Let p be the population proportion  of parents who had children in grades​ K-12 were satisfied with the quality of education the students receive.

Given : Several years​ ago, 39​% of parents who had children in grades​ K-12 were satisfied with the quality of education the students receive.

Set hypothesis to test :

$H_0: p=0.39\\\\H_a :p\neq0.39$

Sample size : n= 1055

Sample proportion : $\hat{p}=\dfrac{466}{1055}=0.441706161137\approx0.44$

Critical value for 95% confidence : $z_{\alpha/2}=1.96$

Confidence interval : $\hat{p}\pm z_{\alpha/2}\sqrt{\dfrac{p(1-p)}{n}}$

$0.44\pm (1.96)\sqrt{\dfrac{0.39(1-0.39)}{1055}}\\\\0.44\pm0.0299536805135\\\\0.44\pm0.03\\\\=(0.44-0.03, 0.44+0.03)\\\\=(0.41,\ 0.47)$

Since , Confidence interval does not contain 0.39.

It means we reject the null hypothesis.

We conclude that 95​% confidence interval represents evidence that​ parents' attitudes toward the quality of education have changed.