Answer:
B) 99% confidence interval is (0.485,0.715)
Step-by-step explanation:
We are given a 90% confidence interval constructed for a population proportion using a sample size of 120. We have to construct a 99% confidence interval from this data. For this first we need the value of sample proportion i.e. the point estimate.
The 90% confidence interval is: (0.526, 0.674). Remember that the point estimate is at an equal distance from both upper limit and lower limit of the confidence interval. So finding out the average of both the limits will give us their middle value i.e. the point estimate or the value of sample proportion.
So, the value of sample proportion will be:
[tex]p=\frac{0.526+0.674}{2}=0.60[/tex]
Now we can calculate the 99% confidence interval using the formula:
[tex](p-z_{\frac{\alpha}{2} } \times \sqrt{\frac{p(1-p)}{n} }, p-z_{\frac{\alpha}{2} } \times \sqrt{\frac{p(1-p)}{n} })[/tex]
Here, [tex]z_{\frac{\alpha}{2}[/tex] is the critical z value for given confidence level. For 99% confidence level, the z value from z-table comes out to be 2.576. The sample size n is given to be 120. Using all these values, we get:
[tex](0.60-2.576 \times \sqrt{\frac{0.60(1-0.60)}{120}}, 0.60+2.576 \times \sqrt{\frac{0.60(1-0.60)}{120}})\\\\ =(0.485,0.715)[/tex]
Therefore, 99% confidence interval is (0.485,0.715)
The 99% confidence interval of the sample proportion obtained for the 95% confidence interval is (0.48; 0.715)
From the interval given :
Standard deviation of proportion :
Hence, the standard deviation, σ is :
Z* = The Critical value at 99% confidence interval = 2.576
The confidence interval can be calculated thus :
Lower boundary = (0.60 - 2.576(0.0447)) = 0.4848
Upper boundary = (0.60 - 2.576(0.0447)) = 0.715
Therefore, the confidence interval is (0.485 ; 0.715)
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