Answer:
[tex]0.58 - 1.64\sqrt{\frac{0.58(1-0.58)}{50}}=0.466[/tex]
[tex]0.58 + 1.64\sqrt{\frac{0.58(1-0.58)}{50}}=0.694[/tex]
The 90% confidence interval would be given by (0.466;0.694)
Step-by-step explanation:
The estimated proportion of interest would be:
[tex] \hat p=\frac{29}{50}= 0.58[/tex]
In order to find the critical value we need to take in count that we are finding the interval for a proportion, so on this case we need to use the z distribution. Since our interval is at 90% of confidence, our significance level would be given by [tex]\alpha=1-0.90=0.1[/tex] and [tex]\alpha/2 =0.05[/tex]. And the critical value would be given by:
[tex]z_{\alpha/2}=-1.64, z_{1-\alpha/2}=1.64[/tex]
The confidence interval for the mean is given by the following formula:
[tex]\hat p \pm z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}}[/tex]
If we replace the values obtained we got:
[tex]0.58 - 1.64\sqrt{\frac{0.58(1-0.58)}{50}}=0.466[/tex]
[tex]0.58 + 1.64\sqrt{\frac{0.58(1-0.58)}{50}}=0.694[/tex]
The 90% confidence interval would be given by (0.466;0.694)
