Question

Doing Time. The U.S. Department of Justice, Office of Justice Programs, and Bureau of Justice Statistics provide information on prison sentences in the document National Corrections Reporting Program. A random sample of $20$ maximum sentences for murder yielded the data, in months, presented on the Weiss Stats site. Use the technology of your choice to do the following.

a. Find a $95\%$ confidence interval for the mean maximum sentence of all murders. Assume a population standard deviation of $30$ months.

b. Obtain a normal probability plot, boxplot, histogram, and stem and leaf diagram of the data.

c. Remove the outliers (if any) from the data, and then repeat part (a).

d. Comment on the advisability of using the $z-$interval procedure on these data.

Short answer

Part (a) The $95\%$confidence interval for the mean maximum sentence of all murders is$(276.802,303.098)$

Part (b)

Stem-and-leaf of TIME $\mathrm{N}=20$

Leaf Unit $=1.0$

$$\begin{gathered} \begin{array}{lll}2& 25& 45\\ 6& 26& 0779\\ 9& 27& 235\\ 10& 28& 7\\ 10& 29& 0125\end{array} \\ 53022 \\ 331 \\ 332 \\ 3333 \\ 2344 \end{gathered}$$

$HI374$

Part (c) $(272.037,299.016)$

Part (d) It is clear that the $z-$interval procedure is not appropriate for the given data.

Step-by-step solution

Part (a) Step 1: Given information

The data on the Weiss Stats site was derived from a random sample of $20$ maximum sentences for murder.

Part (a) Step 2: Explanation

Using MINITAB, calculate the $95\%$confidence interval for the mean maximum sentence of all murders.

MINITAB procedure:

Step 1: Choose Stat $>$Basic Statistics $>$$1-$Sample Z.

Step 2: In Samples in Column, enter the column of TIME.

Step 3: In Standard deviation, enter $30$

Step 4: Select Options and set the Confidence Level to $95$

Step 5: Choose not equal in alternative.

Step 6: Click OK in all dialogue boxes.

MINITAB output:

One-Sample $Z$: TIME

The assumed standard deviation $=30$

From the MINITAB output, the $95\%$confidence interval for the mean maximum sentence of all murders is $(276.802,303.098)$

Part (b) Step 1: Explanation

Using MINITAB, create a boxplot for a maximum sentence for all murders.

MINITAB procedures:

Step 1: Select Graph Boxplot or Stat EDA Boxplot from the menu bar.

Step 2: Select Simple under One $Y\text{'}s$ Click OK.

Step 3: Using TIME data fill in the graph variables.

Step 4: Choose the option OK.

MINITAB PRODUCTIVE OUTPUT:

Draw the normal probability plot for the maximum punishment for all murders using MINITAB.

Procedure for MINITAB:

Step 1: Select Probability Plot from the Graph menu.

Step 2: Click OK after selecting Single.

Step 3: In the Graph variables section, type TIME in the column.

Step 4: Click the OK button.

Part (b) Step 2: Explanation

Draw the histogram for the maximum punishment for all murders using MINITAB.

Procedure for MINITAB:

Step 1: Select Histogram from the Graph menu.

Step 2: Click OK after selecting Simple.

Step 3: In the Graph variables section, type TIME in the relevant column.

Step 4: Click the OK button.

OUTPUT FROM MINITAB:

Draw the stem-and-leaf diagram for the maximum penalty for all murders using MINITAB.

Procedure for MINITAB:

Select Graph $>$ Stem and leaf in the first step.

Step 2: Select TIME as the variable column in Graph variables.

Step 3: Choose OK.

OUTPUT FROM MINITAB:

Stem-and-Leaf Display: TIME

Stem-and-leaf of TIME $\mathrm{N}=20$

Leaf Unit $=1.0$

$$\begin{gathered} \begin{array}{lll}2& 25& 45\\ 6& 26& 0779\\ 9& 27& 235\\ 10& 28& 7\\ 10& 29& 0125\end{array} \\ 53022 \\ 331 \\ 332 \\ 3333 \\ 2344 \end{gathered}$$

$HI374$

Part (c) Step 1: Explanation

Using MINITAB, calculate the $95\%$confidence interval for the mean maximum sentence of all murders after removing outliers.

Procedure for MINITAB:

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

Step 2: In the Samples in Column field, type TIME.

Step 3: Enter $30$in Standard deviation.

Step 4: Select Options and set the Confidence Level to $95$

Step 5: In the alternative, select not equal.

Step 6: In all dialogue boxes, click OK.

OUTPUT FROM MINITAB:

One-Sample $Z$: TIME

The assumed standard deviation $=30$

$\begin{array}{lrrrrc}\text{Variable}& \mathrm{N}& \text{Mean}& \text{StDev}& \text{SE Mean}& 95\text{웅 CI}\\ \text{TIME}& 19& 285.526& 24.174& 6.882& (272.037,299.016)\end{array}$

The $95\%$ confidence interval for the mean maximum sentence of all murders calculated using MINITAB is $(272.037,299.016)$

Part (d) Step 1: Explanation

The results show that there is one significant outlier in the data. With a small sample size, the data distribution seems to be biassed to the right. As a result, the $z-$interval technique is clearly inappropriate for the data.