Question

In a random sample of 37 criminals convicted of a certain​ crime, it was determined that the mean length of sentencing was 64 ​months, with a standard deviation of 10 months. Construct and interpret a 95​% confidence interval for the mean length of sentencing for this crime.

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.)

A. There is a 95​% probability that the mean length of sentencing for the crime is between __ and __ months.

B. We can be 95​% confident that the mean length of sentencing for the crime is between __ and __ months.

C. 95​% of the sentences for the crime are between __ and __ months.

Answer 1

Solution :

Given that,

Point estimate = sample mean = = 64 ​months

sample standard deviation = s = 10 months

sample size = n = 37

Degrees of freedom = df = n - 1 = 37 - 1 = 36

At 95% confidence level

= 1 - 95%  

= 1 - 0.95 =0.05

/2 = 0.025

t/2,df = 2.028

Margin of error = E = t/2,df * (s /n)

= 2.028 * ( 10/ 37)

Margin of error = E = 3.33

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

  ± E

64  ± 3.33

(60.67 , 67.33)

Correct option is B. We can be 95​% confident that the mean length of sentencing for the crime is between 60.67and 67.33 months