Question

Explaining confidence The admissions director for a university found that $(107.8,116.2)$is a $95\%$confidence interval for the mean IQ score of all freshmen. Discuss whether each of the following explanations is correct, based on that information.

a. There is a $95\%$probability that the interval from role="math" localid="1654200953396" $107.8\mathrm{to}116.2$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. If we take many samples, about $95\%$of them 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.

Short answer

a. The explanation of a is incorrect.

b. The explanation of b is incorrect.

c. The explanation of c is correct.

d. The explanation of d is incorrect.

e. The explanation of e is correct.

Step-by-step solution

Given Information

$95\%$ confidence interval for mean IQ is given as$(107.8,116.2)$

To determine whether there is a 95% probability for the interval from 107.8 to 116.2 contains μ

Confidence Interval may or may not contain mean $\mu$.

Probability will be either $0\mathrm{or}1$.

It cannot be $0.95$.

Explanation is incorrect.

To check if there is 95% chance for the interval (107.8,116.2) contains x¯

Sample mean is generally center of given confidence interval.

Probability is $1(\mathrm{not}0.95)$

Explanation is Incorrect.

To check is this interval was constructed using a method that produces intervals that capture the true mean in 95% of all possible samples.

True Mean is also called Population Mean.

Here, interval of $95\%$of possible samples contain population/true mean.

Hence, Explanation is correct.

To check  if about 95% of the samples taken will contain the interval (107.8,116.2)

If sample mean is samples is same, sample contains only same confidence interval. Sample mean in a lot is identical less than $95\%$of possible samples.

Explanation is Incorrect.

To check if the probability for the interval (107.8,116.2) captures μ is either 0 or 1 but it is unknown.

Population mean may or may not contain $\mu$.

Probability is $0\mathrm{or}1$.

Explanation is correct.