Is your favorite TV program often interrupted by advertising? CNBC presented statistics on the average number of programming minutes in a half-hour sitcom (CNBC, February 23,2006)\(.\) The following data (in minutes) are representative of its findings. $$\begin{array}{lll} 21.06 & 22.24 & 20.62 \\ 21.66 & 21.23 & 23.86 \\ 23.82 & 20.30 & 21.52 \\ 21.52 & 21.91 & 23.14 \\ 20.02 & 22.20 & 21.20 \\ 22.37 & 22.19 & 22.34 \\ 23.36 & 23.44 & \end{array}$$ Assume the population is approximately normal. Provide a point estimate and a \(95 \%\) confidence interval for the mean number of programming minutes during a half-hour television sitcom.

To calculate the sample mean, we will add up all the data values and divide by the sample size, n. Let's denote the sum of the data values as \(S\). The sample size, n, is the number of values provided in the dataset. We have 5 rows and 3 columns, with the last value missing, so n = 14. $$ \text{Sample Mean} = \frac{S}{n} $$

Next, we need to find the sample standard deviation. This can be found using the following formula: $$ s=\sqrt{\frac{\Sigma(x_i-\bar{x})^2}{n-1}} $$ where \(x_i\) are the individual data values and \(\bar{x}\) is the sample mean we calculated in step 1.

To calculate the margin of error for the 95% confidence interval, we will need to use the t-distribution critical value because we don't know the population standard deviation. For a 95% confidence interval, the t-distribution critical value can be found using the following equation: $$ t_{\alpha / 2} = t_{0.025} \text{ with } df = n - 1 $$ where \(df\) stands for degrees of freedom, which is equal to the sample size (n) minus 1. We can look up the t-distribution critical value in a t-table or use a calculator that provides it. Now that we have the t-distribution critical value, we can calculate the margin of error using the following formula: $$ \text{Margin of Error} = t_{\alpha / 2} \times \frac{s}{\sqrt {n}} $$ where s is the sample standard deviation we calculated in step 2 and n is the sample size.

Finally, we can now determine the 95% confidence interval for the mean number of programming minutes during a half-hour television sitcom using the point estimate (sample mean) and the margin of error we derived: $$ \text{95% Confidence Interval} = \text{Sample Mean} \pm \text{Margin of Error} $$ Calculate the two values and that will give us our 95% confidence interval for the mean number of programming minutes during a half-hour television sitcom.