Therefore,The point estimate of the population mean (¯x) is 13.The sample standard deviation (s) is approximately 3.391 .The margin of error for the estimation of the population mean is approximately 2.3.The 95% confidence interval for the population mean is approximately [10.7, 15.3].
Certainly! To solve this problem, follow these steps:
Given data:
Sample data from a normal population: 13, 11, 15, 18, 16, 14, 9, 8
(a) Point Estimate of the Population Mean (¯x):
Calculate the sample mean (¯x):
Add all values and divide by the number of values (n):
(13+11+15+18+16+14+9+8/8)
=104/8
=13
The point estimate of the population mean (¯x) is 13.
(b) Point Estimate of the Population Standard Deviation (s):
Calculate the sample standard deviation (s):
Use a formula or calculator to compute the sample standard deviation.
For this dataset, the sample standard deviation (s) is approximately 3.391 (rounded to three decimal places).
(c) Margin of Error for Estimation of Population Mean:
Calculate the Margin of Error (MOE) for a 95% confidence level**:
MOE = Critical value × (s / √n), where n is the sample size.
For a 95% confidence level, the critical value (Z) is approximately 1.96.
MOE=1.96×(3.391/√8)
=1.96×(3.391/2.828)
≈2.343
Therefore, the margin of error for the estimation of the population mean is approximately 2.3 (rounded to one decimal place).
(d) 95% Confidence Interval for Population Mean:
Calculate the 95% Confidence Interval (CI) for the population mean**:
CI=Sample Mean±Margin of Error
CI=13±2.3
The 95% confidence interval for the population mean is approximately [10.7, 15.3].
This provides the point estimates for the population mean and standard deviation, the margin of error for estimation, and the 95% confidence interval for the population mean using the provided sample data.