Question

A husband and wife, Mike and Lori, share a digital music player that has a feature that randomly selects which song to play. Lori claims that Mike loaded more songs than she did. Suppose that when the player was in the random-selection mode, 33 of the first 50 songs selected were songs loaded by Mike. Lori and Mike then construct a 95% confidence interval for the proportion of songs loaded by Mike. The 95% confidence interval is (0.529, 0.791)

1. Interpret a 95% percent confidence interval for the proportion of songs on the player that were loaded by Mike.

2.  The confidence interval (0.529, 0.791) can be referred to as a range of plausible values. For what is this the range of plausible values?

3. When we say we have constructed a 95% confidence interval, what do we expect happens 95% of the time?

4. If Mike and Lori calculated a 99% confidence interval instead of a 95% confidence interval with the same sample (33 out of 50 songs belong to Mike), what effect would the change in confidence level have on the confidence interval?

1. Nothing, the interval will not change.
2. The interval will increase in width.
3. The interval will decrease in width.

Please provide thorough explanations.

Answer 1

1. Interpret a 95% percent confidence interval for the proportion of songs on the player that were loaded by Mike.

We are 95% confident that the proportion of songs on the player that was loaded by Mike will lie between 0.529 and 0.791.

2.  The confidence interval (0.529, 0.791) can be referred to as a range of plausible values. For what is this the range of plausible values?

This range is for the proportion of songs on the player that was loaded by Mike.

3. When we say we have constructed a 95% confidence interval, what do we expect happens 95% of the time?

We expect that the confidence interval will contain the true proportion of songs on the player that was loaded by Mike for 95% of the time.

4. If Mike and Lori calculated a 99% confidence interval instead of a 95% confidence interval with the same sample (33 out of 50 songs belong to Mike), what effect would the change in confidence level have on the confidence interval?

The interval will increase in width.

There is a very clear principle which states when the confidence interval increases, the width of confidence interval increases because it will contain the more amount of the data.