Questions: Listed below are the amounts of net worth (in millions of dollars) of the ten wealthiest celebrities in a country. Consider this as a random sample from a larger population. 186, 196, 180, 165, 162, 154, 151, 154, 149 What is the confidence interval estimate of the population mean μ?

Solution Steps

The sample mean \( \bar{x} \) is calculated as follows:

\[ \bar{x} = \frac{\sum_{i=1}^N x_i}{N} = \frac{1497}{9} = 166.33 \]

The sample variance \( s^2 \) is given by:

\[ s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} = 289.25 \]

The sample standard deviation \( s \) is then:

\[ s = \sqrt{289.25} = 17.01 \]

To calculate the confidence interval for the population mean \( \mu \) at a 95% confidence level, we use the formula:

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

Where:

• \( \bar{x} = 166.33 \)
• \( t \) (for \( n-1 = 8 \) degrees of freedom) is approximately \( 2.31 \)
• \( s = 17.01 \)
• \( n = 9 \)

Substituting the values, we have:

\[ 166.33 \pm 2.31 \cdot \frac{17.01}{\sqrt{9}} = 166.33 \pm 2.31 \cdot \frac{17.01}{3} \]

Calculating the margin of error:

\[ 2.31 \cdot \frac{17.01}{3} \approx 2.31 \cdot 5.67 \approx 13.09 \]

Thus, the confidence interval is:

\[ (166.33 - 13.09, 166.33 + 13.09) = (153.25, 179.41) \]

Final Answer