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

Solution Steps

The mean number of farms per state is calculated using the formula:

\[ \mu = \frac{\sum_{i=1}^N x_i}{N} = \frac{742}{16} = 46.4 \]

Thus, the mean value is:

\[ \text{Mean value} = 46.4 \]

To find the margin of error, we use the formula:

\[ \text{Margin of Error} = \frac{Z \times \sigma}{\sqrt{n}} \]

Given that the Z-score \( Z = 2.2 \), the population standard deviation \( \sigma = 31 \), and the sample size \( n = 16 \), we can substitute these values:

\[ \text{Margin of Error} = \frac{2.2 \times 31}{\sqrt{16}} = 16.8 \]

The 97% confidence interval is calculated as follows:

\[ \text{Lower Bound} = \mu - \text{Margin of Error} = 46.4 - 16.8 = 29.6 \]

\[ \text{Upper Bound} = \mu + \text{Margin of Error} = 46.4 + 16.8 = 63.2 \]

Thus, the 97% confidence interval is:

\[ \text{Confidence Interval} = (29.6, 63.2) \]

Final Answer