Question

In a random sample of 40 felons convicted of aggravated assault, it was determined that the
mean length of sentencing was 54 months, with a standard deviation of 8 months. Construct
and interpret a 95% confidence interval for the mean length of sentence for an aggravated
assault conviction.

Answer 1

Answer on Question #41885 – Math - Statistics and Probability

In a random sample of 40 felons convicted of aggravated assault, it was determined that the mean length of sentencing was 54 months, with a standard deviation of 8 months. Construct and interpret a 95% confidence interval for the mean length of sentence for an aggravated assault conviction.

Solution:

Sample size n = 40 (large sample)

Sample mean $\bar{x} = 54$

Standard deviation $\sigma = 8$

$$\gamma = 0.95 - \text{confidence probability}$$

Since $1 - \alpha = 0.95$ , $\alpha / 2 = 0.025$ , and $z_{0.025} = 1.96$ , the large sample $95\%$ confidence interval for $\mu$ becomes $\left(\bar{x} - z_2 \frac{\sigma}{\sqrt{n}}; \bar{x} - z_2 \frac{\sigma}{\sqrt{n}}\right) = \left(54 - 1.96 \frac{8}{\sqrt{40}}; 54 + 1.96 \frac{8}{\sqrt{40}}\right) = (51.52; 56.48)$ .

This means that we can be 95% confident that the mean length of sentencing is in the interval 51.52 to 56.48 months.

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