Answer:
[tex]CI_{90\%}=[0.6123,\ 0.6465]\\\\CI_{95\%}=[0.6090,\ 0.6498][/tex]
The 95% width is wider
Step-by-step explanation:
-We first calculate the sample proportion,[tex]\hat p[/tex] as follows:
[tex]\hat p=\frac{x}{n}\\\\=\frac{1352}{2148}\\\\=0.6294[/tex]
#The 90% confidence interval can be calculated as:
[tex]CI=\hat p\pm z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}\\\\=0.6294\pm 1.645\times \sqrt{\frac{0.6294(1-0.6294)}{2148}}\\\\=0.6294\pm0.0171\\\\=[0.6123,\ 0.6465][/tex]
#The confidence interval at 95% is calculated as follows:
[tex]CI=\hat p\pm z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}\\\\=0.6294\pm 1.96\times \sqrt{\frac{0.6294(1-0.6294)}{2148}}\\\\=0.6294\pm0.0204\\\\=[0.6090,\ 0.6498][/tex]
-The confidence interval is wider when a higher confidence level and narrower when a lower confidence level is used:
[tex]\bigtriangleup CI_{90\%}=0.6465-0.6123=0.0342\\\\\bigtriangleup CI_{95\%}=0.6498-0.6090=0.0408[/tex]
Hence, the 95% width is wider.
