Question

Millionaires. From Problem 19. we know that "a $95\%$ confidence interval for the mean age of all U.S. millionaires is from $54.3$ years to $62.8$ years." Decide which of the following sentences provides a correct interpretation of the statement in quotes. Justify your answers.

a. Ninety-five percent of all U.S. millionaires are between the ages of $54.3$

years and $62.8$ years.

b. There is a $95\%$ chance that the mean age of all U.S. millionaires is between $54.3$ years and $62.8$ years.

c. We can be $95\%$ confident that the mean age of all U.S. millionaires is between $54.3$ years and $62.8$ years.

d. The probability is $0.95$ that the mean age of all U.S. millionaires is between $54.3$ years and $62.8$ years.

Short answer

Part (a) Wrong interpretations.

Part (b) Wrong interpretations.

Part (c) Correct interpretations.

Part (d) Wrong interpretations.

Step-by-step solution

Part (a) Step 1: Given information

The $95\%$ confidence interval for the mean age of all millionaires in the United States is $54.3$ to $62.8$

Part (a) Step 2: Concept

The formula used:$\left(\overline{x}-z_{\frac{\alpha }{2}}\frac{\sigma }{\sqrt{n}},\overline{x}+z_{\frac{\alpha }{2}}\frac{\sigma }{\sqrt{n}}\right)$

Part (c) Step 1: Explanation

Part (C) is the correct interpretation of the given statement, because a $95\%$confidence interval of $\mu$means that if a large number of samples of the same size are obtained from a population with a mean of $\mu$, and if the interval $\left(\overline{x}-z_{\frac{\alpha }{2}}\frac{\sigma }{\sqrt{n}},\overline{x}+z_{\frac{\alpha }{2}}\frac{\sigma }{\sqrt{n}}\right)$is used for each sample or $\left(\overline{x}-t_{\frac{\alpha }{2}}\frac{s}{\sqrt{n}},\overline{x}+t_{\frac{\alpha }{2}}\frac{s}{\sqrt{n}}\right)$When is found, the interval will encompass $\mu$in around $95\%$ of the situations. That is, we are $95\%$ certain that the obtained (confidence) interval will include $\mu$ for a given sample.

Part (a) Step 3: Explanation

Because a $95\%$ confidence interval of $\mu$ does not indicate that $95\%$ of the population units are inside that interval, portion (a) is incorrect.

Part (b and d) Step 1: Explanation

Both parts (b) and (d) are incorrect interpretations. Because the likelihood (or chance) that the random interval $\left(\overline{x}-z_{\frac{\alpha }{2}}\frac{\sigma }{\sqrt{n}},\overline{x}+z_{\frac{\alpha }{2}}\frac{\sigma }{\sqrt{n}}\right)$will include $\mu$is $0.95(95\%)$An obtained confidence interval is a specific observation of the aforesaid random interval for a given sample. That is, the supplied $(54.3,62.8)$confidence interval of $\mu$is an observed (i.e. constant) interval. In addition, $\mu$is a constant population parameter. As a result, claiming that the chance of a constant interval containing a fixed parameter $\mu$is $0.95(or95\%)$is ridiculous, because probability can only be connected with random occurrences.