Final answer:
To determine the correct 90% confidence interval for the mean sentencing, calculate the margin of error using the z-score for a 90% confidence level and the sample standard deviation, then subtract and add that value from the sample mean. Interpretation means that we are 90% confident that the true mean sentencing length is within the constructed interval.
Step-by-step explanation:
To construct and interpret a 90% confidence interval for the mean length of sentencing for a crime based on a sample mean of 54 months and a standard deviation of 10 months from a sample of 40 criminals, we need to use the z-distribution since the sample size is sufficiently large. Here's how to calculate it:
- Identify the critical value (z*) for a 90% confidence level. For a 90% confidence interval, the critical value is approximately 1.645.
- Calculate the standard error (SE) using the formula SE = s / sqrt(n), where 's' is the sample standard deviation and 'n' is the sample size. In this case, SE = 10 / sqrt(40).
- Calculate the margin of error (ME) as ME = z* × SE.
- The confidence interval is then (mean - ME, mean + ME).
Performing these calculations gives us the margin of error and the confidence interval. To find out which of the provided options is correct, follow these steps with the given sample statistics.
Interpreting the confidence interval means that we are 90% confident that the interval constructed includes the true population mean sentencing length for this crime. This does not mean that the true mean is in the interval with 90% probability; rather, if we were to repeat this process many times, we expect that 90% of the constructed intervals would contain the true mean.