Politicians are interested in knowing the opinions of their constituents on important issues. One administrative assistant to a senator claims that more than 63% of adult women favor stricter gun laws. A recent telephone survey of 1026 adult women by IBR Polls found that 65.9% of adult women favored stricter gun laws.
The 90% confidence interval is: ( % , %)
Enter each number as a percent rounded to tenths of a percent.
Formula:
CI90 = p + or - (1.645)[√(pq/n)]
...where p = x/n, q = 1 - p, and n = sample size.
Note: + or - 1.645 represents 90% confidence interval.
p = .659
q = 1 - p = .341
n = 1026
I let you take it from here.
Would the interval be 63.5 to 68.3?
To calculate the 90% confidence interval, we can use the formula:
\[ \text{Confidence Interval} = \text{Sample proportion} \pm \text{Margin of error} \]
First, let's calculate the sample proportion:
Sample proportion (\( \hat{p} \)) = 65.9% = 0.659
Next, we need to calculate the margin of error, which takes into account the sample size and the desired level of confidence. The formula for the margin of error is:
\[ \text{Margin of error} = \text{Critical value} \times \text{Standard error} \]
The critical value depends on the desired level of confidence and the sample size. For a 90% confidence level, we can find the critical value using a table or a calculator. Since the sample size is 1026, we can assume it is large enough to approximate the critical value as 1.645.
Now, we need to calculate the standard error, which is the standard deviation of the sample proportion. The formula for the standard error is:
\[ \text{Standard error} = \sqrt{\frac{\text{Sample proportion} \times (1 - \text{Sample proportion})}{\text{Sample size}}} \]
Plugging in the values:
Standard error (\(SE\)) = \(\sqrt{\frac{0.659 \times (1 - 0.659)}{1026}}\)
Calculating this equation, we find that the standard error is approximately 0.0137.
Now that we have the critical value and the standard error, we can calculate the margin of error as:
Margin of error = 1.645 \times 0.0137
Calculating this equation, we find that the margin of error is approximately 0.0226.
Finally, we can calculate the confidence interval as:
Confidence Interval = Sample proportion ± Margin of error
Confidence Interval = 0.659 ± 0.0226
Rounding each number to tenths of a percent, the 90% confidence interval is (64.0%, 67.8%).