Questions: 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 97% confidence, Assume σ=31, Use a graphing calculator and round the answers to one decimal place. Assume the population is normally distributed,
79 80 48 16 68 7 15 21 52 6 78 109 40 50 29 44
Transcript text: 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 $97 \%$ confidence, Assume $\sigma=31$, Use a graphing calculator and round the answers to one decimal place. Assume the population is normally distributed,
\[
\begin{array}{llllllllll}
79 & 80 & 48 & 16 & 68 & 7 & 15 & 21 & 52 & 6 \\
78 & 109 & 40 & 50 & 29 & 44 & & & &
\end{array}
\]
Solution
Solution Steps
Step 1: Calculate the Mean
The mean number of farms per state is calculated using the formula:
μ=N∑i=1Nxi=16742=46.4
Thus, the mean value is:
Mean value=46.4
Step 2: Calculate the Margin of Error
To find the margin of error, we use the formula:
Margin of Error=nZ×σ
Given that the Z-score Z=2.2, the population standard deviation σ=31, and the sample size n=16, we can substitute these values:
Margin of Error=162.2×31=16.8
Step 3: Calculate the Confidence Interval
The 97% confidence interval is calculated as follows:
Lower Bound=μ−Margin of Error=46.4−16.8=29.6
Upper Bound=μ+Margin of Error=46.4+16.8=63.2
Thus, the 97% confidence interval is:
Confidence Interval=(29.6,63.2)
Final Answer
The mean number of farms per state is μ=46.4, the margin of error is 16.8, and the 97% confidence interval is (29.6,63.2).