18. Explaining confidence The admissions director from Big City University found that (107.8, 116.2) is a 95% confidence interval for the mean IQ score of all freshmen. Comment on whether or not each of the following explanations is correct.
(a) There is a $95\%$probability that the interval from $107.8$to $116.2$contains $\mu$.
(b) There is a $95\%$chance that the interval $(107.8,116.2$) contains $\overline{x}$.
(c) This interval was constructed using a method that produces intervals that capture the true mean in $95\%$ of all possible samples.
(d) $95\%$of all possible samples will contain the interval $(107.8,116.2)$.

(e) The probability that the interval $(107.8,116.2)$captures $\mu$is either $0$or $1$, but we don’t know which.

To determine that there is a $95\%$probability that the interval from $107.8$to $116.2$ contains $\mu$ is correct or not.

Let, the confidence interval is $95\%$.
And the mean $IQ$is $107.8$to$116.2$.
In the range of $107.8$to $116.2$, the $95$ percent probability does not contain the true population mean $\mu$ of all samples.
As a result, the explanation is incorrect.

To determine that there is a $95\%$chance that the interval $(107.8,116.2)$contains $\overline{x}$is correct or not.

Let, the confidence interval is $95\%$.
And the mean $IQ$is $107.8$to $116.2$.
The probability of finding the sample mean $\overline{x}$ is $95$percent. However, the sample mean $\overline{x}$ is identical to the confidence interval's center.
Hence, the probability is $1$.
As a result, the explanation is incorrect.

To determine that the interval was constructed using a method that
produces intervals that capture the true mean in $95\%$ of all possible samples is correct or not.

Let, the confidence interval is $95\%$.
And the mean $IQ$is $107.8$to $116.2$.
The true mean of all possible samples is included within the $95$ percent confidence interval between $107.8$ and $116.2$.
As a result, the explanation is correct.

To determine that the $95\%$of all possible samples will contain the interval $(107.8,116.2)$is correct or not.

Let, the confidence interval is $95\%$.
And the mean$IQ$is $107.8$to $116.2$.
The interval between $107.8$and $116.2$does not contain real population mean $\mu$in the $95$ percent probability.
As a result, the samples have a confidence interval of less than $95$ percent for all potential samples.
As a result, the explanation is incorrect.

To determine that the probability that the interval $(107.8,116.2)$captures $\mu$is either $0$or $1$, is correct or not.

Let, the confidence interval is $95\%$.
And the mean $IQ$is $107.8$to $116.2$.
The confidence interval from $107.8$to $116.2$ does not contain the true population mean $\mu$ for all samples.

Hence, the probability is either $1$or $1$.
As a a result, the explanation is correct.