Final answer:
To construct a 99% confidence interval for the mean salary of high school counselors in the United States, calculate the sample mean and standard deviation. Use the t-distribution with the appropriate critical value to find the standard error. Then use the formula to calculate the confidence interval.
Step-by-step explanation:
To construct a 99% confidence interval for the mean salary of high school counselors across the United States, we can use the formula:
Confidence Interval = (sample mean) +/- (critical value * standard error)
Step 1: Calculate the sample mean, which is the average of the given salaries: ($61,120 + $52,730 + $53,290 + $36,550 + $34,790 + $61,250 + $61,900) / 7 = $51,601.43 (rounded to the nearest dollar).
Step 2: Calculate the standard deviation of the sample, which is a measure of the variability: $11,037.48 (rounded to the nearest dollar).
Step 3: Calculate the standard error, which is the standard deviation divided by the square root of the sample size. Since this is a small sample size of 7, we can use the t-distribution instead of the z-distribution (since the sample is less than 30). The critical value for a 99% confidence level and 6 degrees of freedom (n-1) is 3.707 (rounded to three decimal places), which we can find from a t-table or a statistical software.
Step 4: Calculate the confidence interval:
Confidence Interval = $51,601.43 +/- (3.707 * ($11,037.48/√7) )
Confidence Interval = $51,601.43 +/- $15,514.87
Confidence Interval ≈ $36,086.56 to $67,116.30 (rounded to the nearest dollar)