Question

1. In a random sample of 8 ​people, the mean commute time to work was 34.534.5 minutes and the standard deviation was 7.27.2 minutes. A 9090​% confidence interval using the​ t-distribution was calculated to be left parenthesis 29.7 comma 39.3 right parenthesis(29.7,39.3). After researching commute times to​ work, it was found that the population standard deviation is 9.19.1 minutes. Find the margin of error and construct a 9090​% confidence interval using the standard normal distribution with the appropriate calculations for a standard deviation that is known. Compare the results. The margin of error of muμ is nothing.

__________ ​(Round to two decimal places as​ needed.)

2. In a random sample of 2525 people, the mean commute time to work was 32.932.9 minutes and the standard deviation was 7.37.3 minutes. Assume the population is normally distributed and use a​ t-distribution to construct a 8080​% confidence interval for the population mean muμ. What is the margin of error of muμ​? Interpret the results. The confidence interval for the population mean muμ is left parenthesis nothing comma nothing right parenthesis .,.

__________​(Round to one decimal place as​ needed.)

PLEASE HELP!!!

Answer 1

Solution:

1) The margin of error for 90% confidence interval of population mean is given by, (when population standard deviation is known)

The 90% confidence interval for population mean (when population standard deviation is known) is given by,

Where, x̄ is sample mean, σ is population standard deviation, n is sample size and Z_(0.10/2) is critical z-value to construct 90% confidence interval.

We have, x̄ = 34.5, σ = 9.1, n = 8

Using Z-table we get, Z_(0.10/2) = 1.645

Hence, margin of error is,

The margin of error is 5.29.

The 90% confidence interval is,

The 90% confidence interval for population mean commute time is (29.21, 39.79).

Comparison: The calculated 90% confidence interval for population mean (when σ known) is wider than the given confidence interval for population mean which is calculated using t distribution.

2) The margin of error for 80% confidence interval of population mean is given by, (using t distribution)

The 80% confidence interval for population mean (using a t distribution) is given by,

Where, x̄ is sample mean, s is sample standard deviation, n is sample size and t_{(0.20/2, n-1)} is critical t-value to construct 80% confidence interval.

We have, x̄ = 32.9, s = 7.3, n = 25

Using t-table we get, t_{(0.20/2, 25 - 1)} = 1.318

Hence, margin of error is,

The margin of error is 1.9.

The 80% confidence interval is,

The 80% confidence interval for population mean commute time is (31.0, 34.8).

Interpretation : We are 80% confident that true value of population mean commute time will lie within the confidence limits of 80% confidence interval.

Note: The calculated values are in minutes.

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