Let’s say you wish to construct a sampling distribution of sample size 100 for the proportion of people that voted for AAP. Suppose the population standard deviation is known to be 0.7, what is the interval in which the mean of the sampling distribution will belong at a 90% confidence level?
Question
Let’s say you wish to construct a sampling distribution of sample size 100 for the proportion of people that voted for AAP. Suppose the population standard deviation is known to be 0.7, what is the interval in which the mean of the sampling distribution will belong at a 90% confidence level?
Solution
To construct a sampling distribution of sample size 100 for the proportion of people that voted for AAP, we need to determine the interval in which the mean of the sampling distribution will belong at a 90% confidence level.
First, we need to calculate the standard error of the sampling distribution. The formula for the standard error of the proportion is:
Standard Error = sqrt((p * (1 - p)) / n)
Where:
- p is the population proportion
- n is the sample size
Since the population standard deviation is known to be 0.7, we can assume that the population proportion is p = 0.5 (assuming a worst-case scenario where the proportion is evenly split). Therefore, the standard error can be calculated as:
Standard Error = sqrt((0.5 * (1 - 0.5)) / 100) = sqrt(0.25 / 100) = 0.05
Next, we need to determine the margin of error for the 90% confidence level. The margin of error is calculated by multiplying the standard error by the critical value from the t-distribution. For a 90% confidence level, the critical value is approximately 1.645.
Margin of Error = 1.645 * 0.05 = 0.08225
Finally, we can calculate the interval in which the mean of the sampling distribution will belong at a 90% confidence level. This is done by subtracting and adding the margin of error to the sample mean.
Interval = Sample Mean ± Margin of Error
Since we don't have the sample mean, we cannot provide the exact interval. However, you can calculate it by substituting the sample mean into the formula.
In summary, to determine the interval in which the mean of the sampling distribution will belong at a 90% confidence level, you need to calculate the standard error, determine the margin of error using the critical value, and then calculate the interval by adding and subtracting the margin of error from the sample mean.
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