Answer: a) Mean and standard deviation of the sampling distribution:
The mean of the sampling distribution, also known as the expected value or the population mean, is equal to the proportion of U.S. residents who receive a jury summons each year, which is 15%. So, the mean of the sampling distribution is:
µ = 0.15
The standard deviation of the sampling distribution, also known as the standard error, can be calculated using the formula:
σ = √(p(1 - p) / n)
where p is the population proportion of U.S. residents who receive a jury summons each year (0.15), and n is the sample size (500).
σ = √(0.15 * (1 - 0.15) / 500)
σ ≈ 0.03
So, the standard deviation of the sampling distribution is approximately equal to 0.03.
b) Interpretation of the standard deviation:
The standard deviation measures the spread of the sampling distribution, or how much the sample proportions deviate from the population proportion. A smaller standard deviation indicates that the sample proportions are more likely to be close to the population proportion, while a larger standard deviation indicates that the sample proportions are more likely to be further from the population proportion.
In this case, the standard deviation of 0.03 indicates that the sample proportions are expected to be within 0.03 of the population proportion (0.15) about 68% of the time.
c) Justification for the approximate normality of the sampling distribution:
According to the central limit theorem, the distribution of the sample proportions will be approximately normal as long as the sample size is large enough. In this case, with a sample size of 500, we can assume that the sampling distribution is approximately normal.
Explanation: