True or false? a. The center of a 95% confidence interval for the population mean is a random variable. b. A 95% confidence interval for μμ contains t

a. $True$. The center of a 95% confidence interval for the population mean is a random variable. This is because the center is calculated based on the sample mean, which is a random variable that can vary from one sample to another.
b. $True$. A 95% confidence interval for $\mu$ contains the sample mean with probability .95. This means that if we were to repeat the sampling and interval construction process multiple times, about 95% of the intervals would contain the true population mean $\mu$.
c. $False$. A 95% confidence interval does not necessarily contain 95% of the population. The 95% confidence level refers to the probability that the interval captures the true population parameter (in this case, the population mean $\mu$), not the proportion of the population contained within the interval.
d. $True$. Out of one hundred 95% confidence intervals for $\mu$, 95 will contain $\mu$. This is consistent with the interpretation of a 95% confidence level.

Step 1:
a. True. The center of a 95% confidence interval for the population mean is a random variable.
In statistics, a confidence interval is constructed based on a random sample from the population. The center of the confidence interval is typically the sample mean, denoted as $\overline{x}$. Since $\overline{x}$ is based on a random sample, it is a random variable itself.
Step 2:
b. True. A 95% confidence interval for $\mu$ contains the sample mean with probability .95.
A 95% confidence interval is constructed such that it has a 95% probability of containing the true population mean, $\mu$. Since the sample mean, $\overline{x}$, is an estimate of $\mu$, the confidence interval will contain $\overline{x}$ with the same probability of 0.95.
Step 3:
c. False. A 95% confidence interval does not necessarily contain 95% of the population.
A confidence interval is a range of values that is constructed to estimate an unknown population parameter, such as the population mean $\mu$. The confidence level, in this case, 95%, represents the probability that the interval will contain the true population parameter. It does not imply that the interval contains a certain percentage of the population.
Step 4:
d. True. Out of one hundred 95% confidence intervals for $\mu$, 95 will contain $\mu$.
If we construct multiple confidence intervals using the same method and with the same confidence level, approximately 95% of those intervals will contain the true population mean, $\mu$. This means that in repeated sampling, about 95 out of 100 intervals will include the true value of $\mu$.

Result:
a)True
b)True
c)False
d)True
Solution:
a. The statement is $\mathbf{\text{True}}$.
In hypothesis testing and confidence interval estimation, we use sample data to make inferences about population parameters. The center of a confidence interval for the population mean is calculated based on the sample mean, which is a random variable. The sample mean is the sum of random variables divided by the sample size. Since the sample mean is a random variable, the center of the confidence interval, which is based on the sample mean, is also a random variable.
b. The statement is $\mathbf{\text{True}}$.
A 95% confidence interval for $\mu$ (population mean) is constructed in such a way that it has a probability of 0.95 of containing the true population mean. This means that if we were to repeat the sampling process and construct 100 different confidence intervals, approximately 95 of them would contain the true population mean. Therefore, a 95% confidence interval for $\mu$ does contain the sample mean with probability 0.95.
c. The statement is $\mathbf{\text{False}}$.
A 95% confidence interval does not guarantee that it contains 95% of the population. The interpretation of a 95% confidence interval is that if we were to repeat the sampling process and construct many confidence intervals, about 95% of them would contain the true population mean. The confidence interval provides information about the precision of our estimate, not about the proportion of the population contained within the interval.
d. The statement is $\mathbf{\text{True}}$.
Based on the definition of a 95% confidence interval, if we were to construct 100 different intervals using the same sampling method, approximately 95 of them would contain the true population mean. The 95% confidence level indicates the level of confidence we have in the procedure of constructing the interval, not the proportion of intervals that will contain the population mean. However, due to the random nature of sampling, the actual number of intervals containing the population mean may vary slightly, but it would be close to 95 out of 100 intervals.