Questions: Listed below are the amounts of net worth (in millions of dollars) of the ten wealthiest celebrities in a country. Construct a 95% confidence interval. What does the result tell us about the population of all celebrities? Do the data appear to be from a normally distributed population as required? 249 202 187 175 162 153 153 153 150 148 150.2 million < mu < 196.2 million (Round to one decimal place as needed.) What does the result tell us about the population of all celebrities? Select the correct choice and, if necessary, fill in the answer box(es) to complete your choice. A. We are 95% confident that the interval from million to million actually contains the true mean net worth of all celebrities. (Round to one decimal place as needed.) B. We are confident that 95% of all celebrities have a net worth between million and million. (Round to one decimal place as needed.) C. Because the ten wealthiest celebrities are not a representative sample, this doesn't provide any information about the population of all celebrities. Do the data appear to be from a normally distributed population as required? Choose the correct answer. A. No, because there is a systematic pattern that is not a straight line pattern. B. No, but the points in the normal quantile plot lie reasonably close to a straight line and show some systematic pattern that is a straight line pattern. C. Yes, but the points in the normal quantile plot do not lie reasonably close to a straight line or show a systematic pattern that is a straight line pattern. D. Yes, because the pattern of the points in the normal quantile plot is reasonably close to a straight line.

Listed below are the amounts of net worth (in millions of dollars) of the ten wealthiest celebrities in a country. Construct a 95% confidence interval. What does the result tell us about the population of all celebrities? Do the data appear to be from a normally distributed population as required? 249 202 187 175 162 153 153 153 150 148 150.2 million < mu < 196.2 million (Round to one decimal place as needed.) What does the result tell us about the population of all celebrities? Select the correct choice and, if necessary, fill in the answer box(es) to complete your choice. A. We are 95% confident that the interval from million to million actually contains the true mean net worth of all celebrities. (Round to one decimal place as needed.) B. We are confident that 95% of all celebrities have a net worth between million and million. (Round to one decimal place as needed.) C. Because the ten wealthiest celebrities are not a representative sample, this doesn't provide any information about the population of all celebrities. Do the data appear to be from a normally distributed population as required? Choose the correct answer. A. No, because there is a systematic pattern that is not a straight line pattern. B. No, but the points in the normal quantile plot lie reasonably close to a straight line and show some systematic pattern that is a straight line pattern. C. Yes, but the points in the normal quantile plot do not lie reasonably close to a straight line or show a systematic pattern that is a straight line pattern. D. Yes, because the pattern of the points in the normal quantile plot is reasonably close to a straight line.
Transcript text: Listed below are the amounts of net worth (in millions of dollars) of the ten wealthiest celebrities in a country. Construct a $95 \%$ confidence interval. What does the result tell us about the population of all celebrities? Do the data appear to be from a normally distributed population as required? 249 202 187 175 162 153 153 153 150 148 $\$ 150.2$ million $<\mu<\$ 196.2$ million (Round to one decimal place as needed.) What does the result tell us about the population of all celebrities? Select the correct choice and, if necessary, fill in the answer box(es) to complete your choice. A. We are $95 \%$ confident that the interval from $\$$ $\square$ million to \$ $\square$ million actually contains the true mean net worth of all celebrities. (Round to one decimal place as needed.) B. We are confident that $95 \%$ of all celebrities have a net worth between $\$$ million and $\$$ million. (Round to one decimal place as needed.) C. Because the ten wealthiest celebrities are not a representative sample, this doesn't provide any information about the population of all celebrities. Do the data appear to be from a normally distributed population as required? Choose the correct answer. A. No, because there is a systematic pattern that is not a straight line pattern. B. No, but the points in the normal quantile plot lie reasonably close to a straight line and show some systematic pattern that is a straight line pattern. C. Yes, but the points in the normal quantile plot do not lie reasonably close to a straight line or show a systematic pattern that is a straight line pattern. D. Yes, because the pattern of the points in the normal quantile plot is reasonably close to a straight line.
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Solution

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Solution Steps

Step 1: Calculate the Mean

The mean net worth of the ten wealthiest celebrities is calculated as follows:

\[ \mu = \frac{\sum_{i=1}^N x_i}{N} = \frac{1732}{10} = 173.2 \]

Step 2: Calculate the Standard Deviation

The variance is calculated using the formula:

\[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n-1} = 1032.4 \]

Thus, the standard deviation is:

\[ \sigma = \sqrt{1032.4} = 32.1 \]

Step 3: Construct the Confidence Interval

To construct the 95% confidence interval for the mean, we use the formula:

\[ \bar{x} \pm t \frac{s}{\sqrt{n}} \]

Where:

  • \(\bar{x} = 173.2\)
  • \(t \approx 2.3\) (for \(n-1 = 9\) degrees of freedom at 95% confidence)
  • \(s = 32.1\)
  • \(n = 10\)

Calculating the confidence interval:

\[ 173.2 \pm 2.3 \cdot \frac{32.1}{\sqrt{10}} = (150.2, 196.2) \]

Step 4: Interpret the Confidence Interval

We are 95% confident that the interval from \(\$150.2\) million to \(\$196.2\) million actually contains the true mean net worth of all celebrities.

Step 5: Assess Normality of the Data

The data appear to be from a normally distributed population because the pattern of the points in the normal quantile plot is reasonably close to a straight line.

Final Answer

The answers to the questions are:

  • A. We are 95% confident that the interval from \(\$150.2\) million to \(\$196.2\) million actually contains the true mean net worth of all celebrities.
  • D. Yes, because the pattern of the points in the normal quantile plot is reasonably close to a straight line.

Thus, the final boxed answers are:

\[ \boxed{A} \]

\[ \boxed{D} \]

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