Questions: A researcher conducted a survey of Florida police departments and found that of the 18 departments sampled (N=18), there was a mean of 13.00 school resource officers (s=12.10). Construct a 95% confidence interval around the sample mean and interpret your results.

A researcher conducted a survey of Florida police departments and found that of the 18 departments sampled (N=18), there was a mean of 13.00 school resource officers (s=12.10). Construct a 95% confidence interval around the sample mean and interpret your results.

Transcript text: 27. A researcher conducted a survey of Florida police departments and found that of the 18 departments sampled ( $\mathrm{N}=18$ ), there was a mean of 13.00 school resource officers ( $\mathrm{s}=12.10$ ). Construct a $95 \%$ confidence interval around the sample mean and interpret your results. (4 points)

Solution

Solution Steps

Step 1: Given Data

The researcher conducted a survey of Florida police departments with the following statistics:

Sample size ($N$): 18
Sample mean ($\bar{x}$): 13.00
Sample standard deviation ($s$): 12.10
Confidence level: 95%

Step 2: Confidence Interval Calculation

To construct the 95% confidence interval for the mean, we use the formula:

\[ \bar{x} \pm t \frac{s}{\sqrt{n}} \]

Where:

$\bar{x} = 13.00$
$t$ is the t-value for $N-1 = 17$ degrees of freedom at a 95% confidence level, which is approximately 2.11.
$s = 12.10$
$n = 18$

Step 3: Margin of Error Calculation

The margin of error is calculated as follows:

\[ \text{Margin of Error} = t \cdot \frac{s}{\sqrt{n}} = 2.11 \cdot \frac{12.10}{\sqrt{18}} \approx 2.11 \cdot 2.85 \approx 6.00 \]

Step 4: Confidence Interval Result

Now, we can calculate the confidence interval:

\[ \text{Confidence Interval} = \left( \bar{x} - \text{Margin of Error}, \bar{x} + \text{Margin of Error} \right) = \left( 13.00 - 6.00, 13.00 + 6.00 \right) = (7.00, 19.00) \]