Questions: Dance Company Students The number of students who belong to the dance company at each of several randomly selected small universities is shown below. Round sample statistics and final answers to at least one decimal place. 28 28 26 25 22 21 47 40 35 32 30 29 26 40 Estimate the true population mean size of a university dance company with 80% confidence. Assume the variable is normally distributed.

Solution Steps

The mean of the number of students in the dance company is calculated as follows:

\[ \mu = \frac{\sum_{i=1}^N x_i}{N} = \frac{429}{14} \approx 30.6 \]

The variance is calculated using the formula:

\[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n-1} \approx 55.63 \]

Thus, the standard deviation is:

\[ \sigma = \sqrt{55.63} \approx 7.46 \]

For an 80% confidence level, the Z-score is approximately \( Z \approx 1.28 \). The margin of error is calculated as:

\[ \text{Margin of Error} = \frac{Z \times \sigma}{\sqrt{n}} = \frac{1.28 \times 7.46}{\sqrt{14}} \approx 2.56 \]

The confidence interval for the population mean \( \mu \) is given by:

\[ \text{Lower Bound} = \mu - \text{Margin of Error} \approx 30.6 - 2.56 \approx 28.04 \] \[ \text{Upper Bound} = \mu + \text{Margin of Error} \approx 30.6 + 2.56 \approx 33.16 \]

Thus, the 80% confidence interval is:

\[ 28.04 < \mu < 33.16 \]

Final Answer