Question

ha random sample of 31 criminals convicted of a certain crime, it was determined that the mean length of sentencing was 54 months, with a standard deviation of 6 months. Construct and interpreta 95% confidence interval for the mean length of sentencing for this crime. Click here to view the standard normal distribution table lenge 1) Click here to view the standard normal distribution table (page 2) Click here to view the table of critical 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) OA. We can be 95% confident that the mean length of sentencing for the crime is between and months OB. 05% of the sentences for the crime are between and months. OC. There is a 95% probability that the mean length of sentencing for the crime is between and months

Answer 1

Solution :

Given that,

$\bar x$ = 54

s = 6

n = 31

Degrees of freedom = df = n - 1 = 31 - 1 = 30

At 95% confidence level the t is ,

$\alpha$  = 1 - 95% = 1 - 0.95 = 0.05

$\alpha$ / 2 = 0.05 / 2 = 0.025

t_{$\alpha$ /2,df} = t_0.025,30 =2.042  

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

= 2.042 * (6 / $\sqrt$ 31)

= 2.2

Margin of error = 2.2

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

$\bar x$ - E <$\mu$  < $\bar x$ + E

54 - 2.2 < $\mu$ < 54 + 2.2

51.80 < $\mu$ < 56.2

We can be 95% confident that the mean lenght of sentencing for the crime in between 51.80 and 56.2 months.