Q 8.21RP. - Elementary Statistics

Question

Prison Sentences. Researchers M. Dhami et al. discussed how people adjust to prison life in the article "Adaption to Imprisonment" (Criminal Justice and Behavior. Vol. $34$, No. $8$, pp. $1085-1100$). A sample of $712$ federally sentenced adult male prisoners had an average sentence of $9.15$ years. Assume that, for federally sentenced adult male prisoners, the population standard deviation of sentence length is $17.2$years.

a. Find and interpret a $90\%$ confidence interval for the mean sentence length, $\mu$, of all federally sentenced adult male prisoners.

b. Under what conditions can you freely apply the procedure that you used in part (a)?

Step-by-Step Solution

Answer

Part (a) The $90\%$ confidence interval for all federally convicted adult male inmates' mean sentence length is $(8.0897,10.2103)$

Part (b) It is clear that the applying z-interval procedure to obtain a confidence interval for the population mean is appropriate.

Part (a) Step 1: Given information

$712$ federally sentenced adult male convicts received an average sentence of $9.15$ years. Assume that the median sentence term for federally sentenced adult male offenders is $17.2$ years.

Part (a) Step 2: Calculation

Using MINITAB, calculate the $90\%$ confidence interval for the average sentence time of all federally sentenced adult male convicts.

Procedure for MINITAB:

Step 1: Select Stat $>$ Basic Statistics $>$ $1-$Sample $Z$ from the drop-down menu.

Step 2: In Summarized Data, enter $712$ as the sample size and $9.15$ as the mean.

Step 3: In the Standard deviation box, enter $17.2$ for $s$

Step 4: Go to Options and set the Confidence Level to $90$

Step 5: Choose the less expensive option.

Step 6: In all dialogue boxes, click OK.

One-Sample $Z$ MINITAB output.

The assumed standard deviation $=17.2$

N	Mean	SE Mean	90% CI
712	9.15000	0.64460	(80.08973, 10.21027)

The $90\%$ confidence interval for the mean sentence length of all federally sentenced adult male convicts is $(8.0897,10.2103)$, according to MINITAB output.

The average sentence term of all federally sentenced adult male Vat prisoner: lies between $8.9897$ and $10.2103$ is $90\%$ certain.

Part (b) Step 1: Calculation

The following are the conditions for using the $z-$interval procedure:

Small Sample size:

The $z-$interval approach is employed when the sample size is less than $15$ and the variable is normally distributed or extremely near to being normally distributed.

Moderate Sample size:

When the sample size is between $15$ and $30$, and the variable is not normally distributed or there is no outlier in the data, the $z-$interval technique is performed.

Large Sample size:

The $z-$interval technique is utilized without restriction if the sample size is bigger than $30$

The sample is drawn from the population, with a huge sample size of $(=712)$

Furthermore, the standard deviation of the population is known. Apart from the shape of the distribution, the sample means distribution is approximately normal. The application of the $z-$interval approach to creating$(=712)$ a confidence interval for the population mean is clearly reasonable given the parameters.