Question

in a random sample of 41 criminals convicted of a certain crime, it was determined that the mean length of sentencing was 65 monghs. with a standard deviation of 7 months. construct and interpret a 95% confidence interval for the mean length of sentencing for this crime.

a) 95% of the sentences for the crime are between ___ and ___ months

b) one can be 95% confident that the mean length if sentencing fir this crime is between __ and __ months

c) there is 95% probability that the mean length of sentencing for the crime is between __ and ___ months

Answer 1

Solution :

Given that,

Point estimate = sample mean = = 65

sample standard deviation = s = 7

sample size = n = 41

Degrees of freedom = df = n - 1 = 41 - 1 = 40

At 95% confidence level the t is ,

= 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

t _/2,df = t_0.025,40 = 2.021

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

= 2.021 * (7 / 41)

= 2.2

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

- E < < + E

65 - 2.2< < 65 + 2.2

62.7 < < 67.2

b)

one can be 95% confident that the mean length if sentencing fir this crime is between 62.7 and 67.2 months .