Question

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

Confidence intervals
ANSWERED
asked 2021-06-02
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 the sample mean with probability .95.
c. A 95% confidence interval contains 95% of the population.
d. Out of one hundred 95% confidence intervals for μμ, 95 will contain μμ.

Answers (1)

2021-06-03
A 95’ confidence interval for the population mean j is of the form:
\(\displaystyle\overline{{{x}}}\pm{\frac{{{z}_{{\alpha}}}}{{{2}}}}{\frac{{\sigma}}{{\sqrt{{{n}}}}}}\)
(a) The center of a, 95% confidence interval is the sample mean Z.
A random variable is a variable whose possible values are numerical outcomes of a random phenomenon.
The sample mean is an outcome of random phenomenon, because it can vary for different samples. Thus the sample mean, the center of the confidence interval, is a random variable.
This then means that the given statement is true.
2.(b) The 95’ confidence interval always contains the sample mean, because it is the center of the confidence interval.
This then means that the confidence interval contains the sample mean with probability 1 (since an event that is known to always occur has probability 1).
Thus the given statement is then false, becatise the probability should instead of 0.95.
3.(c)The 95% confidence interval contains estimates for the population mean, but not for the population itself, thus the given statement is false.
4.(d) 95% of all possible 95’ confidence intervals for » will contain the true population mean \(\displaystyle\mu\).
However, this then does not mean that exactly 95 of 100 confidence intervals will contain the true population mean jz. It could, for example, be 94 confidence intervals or 96 confidence intervals.
Thus the given statement is false.
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