The confidence interval for population mean is given by :-
[tex]\hat{p}\pm z_{\alpha/2}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex]
, where [tex]\hat{p}[/tex] is the sample proportion, n is the sample size , [tex]z_{\alpha/2}[/tex] is the critical z-value.
The values needed to calculate a confidence interval at the 99% confidence level are :
Given : Significance level : [tex]\alpha:1-0.99=0.01[/tex]
Sample size : n=450
Critical value : [tex]z_{\alpha/2}=2.576[/tex]
Sample proportion: [tex]\hat{p}=\dfrac{280}{450}\approx0.62[/tex]
Now, the 99% confidence level will be :
[tex]\hat{p}\pm z_{\alpha/2}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}\\\\=0.62\pm(2.576)\sqrt{\dfrac{0.62(1-0.62)}{450}}\\\\\approx0.62\pm0.023\\\\=(0.62-0.023,\ 0.62+0.023)=(0.597,\ 0.643)[/tex]
Hence, the 99% confidence interval is [tex](0.597,\ 0.643)[/tex]
Answer:
(0.597, 0.643)
Step-by-step explanation:
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