Final answer:
a. The estimated population mean is $65.00 per hour. b. The 90% confidence interval for the population mean wage is approximately $63.24 to $66.76. c. A sample size of at least 110 is needed to assess the population mean with an allowable error of $100 at 98% confidence.
Step-by-step explanation:
a. Estimated population mean:
The best estimate of the population mean is the sample mean, which is $65.00 per hour.
b. 90% confidence interval:
To calculate the confidence interval for the population mean wage, we can use the formula:
CI = X ± (t * (S/sqrt(n)))
Where:
X = Sample mean ($65.00)
t = t-score from the t-distribution table for the desired confidence level (90% confidence corresponds to a t-score of approximately 1.70)
S = Sample standard deviation ($5.72)
n = Sample size (24)
Plugging in the values, we have:
CI = $65.00 ± (1.70 * ($5.72/sqrt(24)))
Calculating this, we find that the 90% confidence interval for the population mean wage is approximately $63.24 to $66.76.
c. Sample size:
To determine the sample size needed to assess the population mean with an allowable error of $100 at 98% confidence, we can use the formula:
n = ((Z * S)/E)^2
Where:
n = Sample size
Z = Z-score from the standard normal distribution table for the desired confidence level (98% confidence corresponds to a Z-score of approximately 2.33)
S = Population standard deviation (given as $5.72 in the sample)
E = Allowable error ($100)
Plugging in the values, we have:
n = ((2.33 * $5.72)/$100)^2
Calculating this, we find that a sample size of at least 110 is needed to assess the population mean with an allowable error of $100 at 98% confidence.