Question

An NHANES report gives data for 654 women aged 20 to 29 years. The mean BMI for these 654 women was x = 26.8. On the basis of this sample, we are going to estimate the mean    BMI μ in the population of all 20.6 million women in this age group. We will assume that the NHANES sample is a SRS from a normal distribution with known standard deviation σ = 7.5

1. Construct three confidence intervals for the mean BMI μ in this population using 90%, 95%, and 99% confidence.
2. What are the margin of errors for 90%, 95%, and 99% confidence?
3. How does increase in confidence level change the margin of error when the sample size and population standard deviation remain the same?
4. Suppose we have a SRS of just 100 young women. What would be the margin of error for 95% confidence?
5. Find the margins of error for 95% confidence based on SRSs of 400 young women.
6. Find the margins of error for 95% confidence based on SRSs of 1600 young women.
7. Compare the three margins of errors. How does increase in sample size change the margin of error when the confidence level and population standard deviation remain the same.

Answer 1

Standard error of mean = = 7.5 / = 0.2932732

Z value for 90%, 95% and 99% confidence interval is 1.645, 1.96 and 2.576 respectively.

Margin of errors for 90%, 95%, and 99% confidence are

1.645 * 0.2932732, 1.96 * 0.2932732 and 2.576 * 0.2932732

0.4824344 , 0.5748155 and 0.7554718

With increase in confidence level, the margin of error increases when the sample size and population standard deviation remain the same.

For n = 100, Standard error of mean = = 7.5 / = 0.75

Margin of error = 1.96 * 0.75 = 1.47

For n = 400, Standard error of mean = = 7.5 / = 0.375

Margin of error = 1.96 * 0.375 = 0.735

For n = 1600, Standard error of mean = = 7.5 / = 0.1875

Margin of error = 1.96 * 0.1875 = 0.3675

The margin of error is highest for n = 100 and lowest for n = 1600.

With increase in sample size, the margin of error decreases when the confidence level and population standard deviation remain the same.