Number of Farms A random sample of the number of farms (in thousands) in various states follows. Estimate the mean number of farms per state with $96 \%$ confidence. Assume $\sigma=31$. Use a graphing calculator and round the answers to one decimal place. Assume the population is normally distributed.
\begin{tabular}{llllllllll}
47 & 95 & 54 & 33 & 64 & 8 & 90 & 3 & 49 & 68 \\
7 & 15 & 21 & 52 & 6 & 78 & 109 & 40 & 50 &
\end{tabular}

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Step 1 ：We are given a sample of data on the number of farms (in thousands) in various states and we are asked to estimate the mean number of farms per state with a 96% confidence level. We are also given that the standard deviation (σ) is 31 and that the population is normally distributed.

Step 2 ：To calculate the confidence interval, we can use the formula for a confidence interval for a population mean: \(CI = \bar{x} ± Z * (σ/\sqrt{n})\), where \(\bar{x}\) is the sample mean, Z is the Z-score (which we can find using a Z-table or a calculator for the given confidence level of 96%), σ is the standard deviation of the population, and n is the sample size.

Step 3 ：First, we need to calculate the sample mean (\(\bar{x}\)) and the sample size (n). The sample data is [47, 95, 54, 33, 64, 8, 90, 3, 49, 68, 7, 15, 21, 52, 6, 78, 109, 40, 50]. The sample mean is approximately 46.8 and the sample size is 19.

Step 4 ：Next, we find the Z-score for a 96% confidence level. The Z-score for a 96% confidence level is approximately 2.05.

Step 5 ：Finally, we can calculate the confidence interval using the formula \(CI = \bar{x} ± Z * (σ/\sqrt{n})\). Substituting the values we have, the lower limit of the confidence interval is approximately 32.2 and the upper limit is approximately 61.4.

Step 6 ：Final Answer: The 96% confidence interval for the mean number of farms per state is \(\boxed{32.2}\) to \(\boxed{61.4}\) (rounded to one decimal place).