Question

Question 1: A researcher would like to estimate the true mean number of hours adults sleep at night. Suppose that population sleep time is known to follow a Normal distribution with standard deviation as 1.5 hours. The researcher random select 100 people and found the average sleeping hours for the sample of 100 people is 6.5 hours.

1) Use this sample mean to estimate the true population mean of sleep time with 95% confidence. (hint: 95% CI)

2) If the researcher intends to increase the confidence of the estimation to 99%, what is the CI now. Compare the 99% CI with the 95% CI calculated in previous question, what has been changed?

3) If the researcher wants to increase the confidence of estimation to 99% without extend the margin of error, what should this researcher do? (hint: sample size)

4) Reflect on previous questions and the CI formula , what factors can affect the CI and how?

Answer 1

1)

Standard error of mean = 1.5 / = 0.15

Z score for 95% confidence interval is 1.96

95% CI of true population mean of sleep time is,

(6.5 - 1.96 * 0.15 ,  6.5 + 1.96 * 0.15)

(6.206 hours , 6.794 hours)

2)

Only Z score for 99% confidence interval is changed.

Z score for 99% confidence interval is 2.576

95% CI of true population mean of sleep time is,

(6.5 - 2.576 * 0.15 ,  6.5 + 2.576 * 0.15)

(6.1136 hours , 6.8864 hours)

99% CI is wider than 95% CI.

3)

To increase the confidence of estimation to 99% without extend the margin of error, we need to increase the sample size. Increasing the sample size will reduce the standard error and margin of error.

4)

CI formula is,

The factors can affect the CI are,

• Confidence level (Increasing the confidence level will increase the width of the CI)
• Sample Size (Increasing the sample size will decrease the width of the CI)
• Margin of error (E = ) Margin of error depends on confidence level and sample size. Increase in the margin of error will increase the width of the CI.