a. The population mean is the average wage (including benefits) of all judges employed by the state of Florida. Since we do not have data on the entire population, we can estimate the population mean using the sample mean. The point estimate of the population mean is the same as the sample mean, which is $67.00 per hour.
b. To develop a 99% confidence interval for the population mean wage (including benefits), we can use the formula:
Confidence Interval = sample mean ± (critical value) * (standard deviation / sqrt(sample size))
The critical value depends on the desired confidence level and the sample size. For a 99% confidence level, the critical value can be obtained from a standard normal distribution table, and it is approximately 2.62 for a sample size of 29 (with a degrees of freedom of 28). The standard deviation is given as $6.17 per hour, and the sample size is 29.
Confidence Interval = $67.00 ± (2.62 * ($6.17 / sqrt(29)))
Calculating this expression will give you the confidence interval for the population mean wage.
c. To estimate the population mean wage with 95% confidence and a margin of $2.00, we need to determine the sample size. The formula to calculate the required sample size is:
n = (z * (sigma / E))^2
Where:
- n is the required sample size
- z is the z-score corresponding to the desired confidence level (95% confidence level corresponds to a z-score of approximately 1.96)
- sigma is the estimated standard deviation
- E is the desired margin of error
In this case, we can use the given standard deviation of $6.17 per hour and the desired margin of $2.00. Plugging these values into the formula will give you the required sample size.