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.

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

Step-by-step 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. =    ~

where, = sample mean length of sentencing = 51 months

              = sample standard deviation = 11 months

             n = sample of criminals = 41

              = 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,  is ;

P(-2.021 < < 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 < < 2.021) = 0.95

P( < < ) = 0.95

P( < < ) = 0.95

95% confidence interval for = [ , ]

                                               = [ , ]

                                              = [47.53 , 54.47]

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

a negative exponent is translated as follows:

so, we have