Question

You may need to use the appropriate appendix table or technology to answer this question. The following sample data are from a normal population: 13, 11, 15, 18, 16, 14, 9, 8, (a) What is the point estimate of the population mean? (b) What is the point estimate of the population standard deviation? (Round your answer to three decimal places.) (c) with 95% confidence, what is the margin of error for the estimation of the population mean? (Round your answer to one decimal place.) (d) What is the 95% confidence interval for the population mean? (Round your answer to one decimal place.) to

Answer 1

a. The point estimate for the population mean is 13. b. The point estimate for the population standard deviation is approximately 3.692. c. The margin of error for the estimation of the population mean with 95% confidence is approximately 3.286. d. The 95% confidence interval for the population mean is (9.714, 16.286).

Step-by-step explanation:

a. The point estimate for the population mean can be calculated by finding the mean of the sample data. Adding up the sample data, we get 13 + 11 + 15 + 18 + 16 + 14 + 9 + 8 = 104. Dividing by the sample size (8), we get a point estimate of 13.

b. The point estimate for the population standard deviation can be calculated by finding the sample standard deviation. Using the formula for sample standard deviation, we find that the point estimate is approximately 3.692.

c. To calculate the margin of error for the estimation of the population mean with 95% confidence, we need to find the appropriate t-value. Since the sample size is small (8), we use a t-distribution. Looking up the t-value for a 95% confidence level and 7 degrees of freedom in the t-table, we find that the t-value is approximately 2.365. To calculate the margin of error, we multiply the t-value by the standard error, which is the standard deviation divided by the square root of the sample size. The margin of error is approximately 2.365 * (3.692 / sqrt(8)) = 3.286.

d. The 95% confidence interval for the population mean can be calculated by subtracting the margin of error from the point estimate and adding it to the point estimate. The lower bound of the confidence interval is 13 - 3.286 = 9.714 and the upper bound is 13 + 3.286 = 16.286. So, the 95% confidence interval for the population mean is (9.714, 16.286).

Answer 2

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.