Question

In a random sample of 41 criminals convicted of a certain​ crime, it was determined that the mean length of sentencing was 51 ​months, with a standard deviation of 11 months. Construct and interpret a 95​% confidence interval for the mean length of sentencing for this crime.

Answer 1

95​% confidence interval for the mean length of sentencing for this crime is [47.53 months , 54.47 months].

Explanation:

We are given that in a random sample of 41 criminals convicted of a certain​ crime, it was determined that the mean length of sentencing was 51 ​months, with a standard deviation of 11 months.

Firstly, the pivotal quantity for 95% confidence interval for the population mean is given by;

P.Q. =
`(\bar X-\mu)/((s)/(√(n) ) )` ~
`t_n_-_1`

where,
`\bar X` = sample mean length of sentencing = 51 months

`s` = sample standard deviation = 11 months

n = sample of criminals = 41

`\mu` = population mean length of sentencing

Here for constructing 95% confidence interval we have used One-sample t test statistics as we don't know about population standard deviation.

So, 95% confidence interval for the population mean,
`\mu` is ;

P(-2.021 <
`t_4_0` < 2.021) = 0.95 {As the critical value of t at 40 degree of

freedom are -2.021 & 2.021 with P = 2.5%}

P(-2.021 <
`(\bar X-\mu)/((s)/(√(n) ) )` < 2.021) = 0.95

P(
`-2.021 * {(s)/(√(n) ) }` <
`{\bar X-\mu}` <
`-2.021 * {(s)/(√(n) ) }` ) = 0.95

P(
`\bar X-2.021 * {(s)/(√(n) ) }` <
`\mu` <
`\bar X+2.021 * {(s)/(√(n) ) }` ) = 0.95

95% confidence interval for
`\mu` = [
`\bar X-2.021 * {(s)/(√(n) ) }` ,
`\bar X+2.021 * {(s)/(√(n) ) }` ]

= [
`51-2.021 * {(11)/(√(41) ) }` ,
`51+2.021 * {(11)/(√(41) ) }` ]

= [47.53 , 54.47]

Therefore, 95​% confidence interval for the mean length of sentencing for this crime is [47.53 months , 54.47 months].