Chapter 8.1, Problem 18E

(a)

To determine

Whether there is a 95% probability for the interval from 107.8 to 116.2 contains $\mu$ .

Answer to Problem 18E

The explanation is incorrect.

Explanation of Solution

Given information:

95% confidence interval for the mean IQ:

(107.8, 116.2)

Since the confidence interval either contains or does not contain the population mean $\mu$ .

Then

We know that the probability is either 0 or 1 (not 0.95).

Thus,

The explanation is incorrect.

(b)

To determine

Whether there is 95% chance for the interval (107.8, 116.2) contains $\overline{x}$ .

Answer to Problem 18E

The explanation is incorrect.

Explanation of Solution

Given information:

95% confidence interval for the mean IQ:

(107.8, 116.2)

Since the confidence interval always contains the sample mean.

Also,

Sample mean is the center of the confidence interval.

And

The probability is 1 instead of 0.95.

Thus,

The explanation is incorrect.

(c)

To determine

Whether the interval was constructed with method used to produce intervals for capturing the true mean in 95% of all possible samples.

Answer to Problem 18E

The explanation is correct.

Explanation of Solution

Given information:

95% confidence interval for the mean IQ:

(107.8, 116.2)

We know that

True mean is also known as population mean.

In this case,

The confidence interval of 95% of all possible samples will contain the true (population) mean.

Thus,

The explanation is correct.

(d)

To determine

Whether about 95% of the samples taken will contain the interval (107.8, 116.2).

Answer to Problem 18E

The explanation is incorrect.

Explanation of Solution

Given information:

95% confidence interval for the mean IQ:

(107.8, 116.2)

Since the samples will only contain the same confidence interval if the sample mean in the samples will be same.

Also,

The sample mean will be identical in a lot less than 95% of all possible samples.

Thus,

The explanation is incorrect.

(e)

To determine

Whether the probability for the interval (107.8, 116.2) captures $\mu$ is either 0 or 1, but it is unknown.

Answer to Problem 18E

The explanation is correct.

Explanation of Solution

Given information:

95% confidence interval for the mean IQ:

(107.8, 116.2)

Since the confidence interval either contains the population mean $\mu$ or not.

Also,

The probability is either 1 or 0 (not 0.95).

Thus,

The explanation is correct.