Question

9.3.11 Question Help In a random sample of 39 criminals convicted of a certain crime, it was determined that the mean length of sentencing was 50 months, with a standard deviation of 11 months. Construct and interpret a 95% confidence interval for the mean length of sentencing for this crime. Click here to view the standard normal distribution table (page 1) Click here to view the standard normal distribution table (page 2) Click here to view the table of critical t-values Select the correct choice below and fill in the answer boxes to complete your choice. (Use ascending order. Round to one decimal place as needed.) O A. We can be 95% confident that the mean length of sentencing for the crime is between 46.9 and 53.1 months. O B. months 95% of the sentences for the crime are between and O C. There is a 95% probability that the mean length of sentencing for the crime is between and months.

Answer 1

Solution :

Given that,

Point estimate = sample mean = $\bar x$ = 50

sample standard deviation = s = 11

sample size = n = 39

Degrees of freedom = df = n - 1 = 39 - 1 = 38

At 95% confidence level

$\alpha$ = 1 - 99%

$\alpha$ =1 - 0.95 =0.05

$\alpha$/2 = 0.025

t$\alpha$/2,df = t_{0.025, 38} = 2.024

Margin of error = E = t_{$\alpha$/2,df} * (s /$\sqrt$n)

= 2.024 * ( 11/ $\sqrt$ 39)

Margin of error = E = 3.6

The 95% confidence interval estimate of the population mean is,

$\bar x$  ± E

= 50  ± 3.6

= ( 46.4, 53.6)

We can be 95% confident that the mean length of sentencing for the crime is between 46.4 and 53.6 months.