Question

Several years​ ago, 38​% of parents who had children in grades​ K-12 were satisfied with the quality of education the students receive. A recent poll asked 1 comma 165 parents who have children in grades​ K-12 if they were satisfied with the quality of education the students receive. Of the 1 comma 165 ​surveyed, 499 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

`0.428 - 1.96\sqrt{(0.428(1-0.428))/(1165)}=0.3995`

`0.428 + 1.96\sqrt{(0.428(1-0.428))/(1165)}=0.4564`

We are confident that the true proportion of people satisfied with the quality of education the students receive is between (0.3995, 0.4564), since the lower value for this confidence level is higher than 0.38 we have enough evidence to conclude that the parents' attitudes toward the quality of education have changed.

Explanation:

For this case we are interesting in the parameter of the true proportion of people satisfied with the quality of education the students receive

The confidence level is given 95%, the significance level would be given by
`\alpha=1-0.95=0.05` and
`\alpha/2 =0.025`. And the critical values are:

`z_(\alpha/2)=-1.96, z_(1-\alpha/2)=1.96`

The estimated proportion of people satisfied with the quality of education the students receive is given by:

`\hat p =(499)/(1165)= 0.428`

The confidence interval for the proportion if interest is given by the following formula:

`\hat p \pm z_(\alpha/2)\sqrt{(\hat p (1-\hat p))/(n)}`

And replacing the info given we got:

`0.428 - 1.96\sqrt{(0.428(1-0.428))/(1165)}=0.3995`

`0.428 + 1.96\sqrt{(0.428(1-0.428))/(1165)}=0.4564`

We are confident that the true proportion of people satisfied with the quality of education the students receive is between (0.3995, 0.4564), since the lower value for this confidence level is higher than 0.38 we have enough evidence to conclude that the parents' attitudes toward the quality of education have changed.