Q 8.140. - Elementary Statistics

Question

Forearm Length. In $1903$, K. Pearson and A. Lee published the paper "On the Laws of Inheritance in Man. I. Inheritance of Physical Characters" (Biometrika, Vol. 2, Pp. 357-462). The article examined and presented data on forearm length, in inches, for a sample of 140 men, which we have provided on the Weiss Stats site. Use the technology of your choice to do the following.

a. Obtain a normal probability plot, boxplot, and histogram of the data.

b. Is it reasonable to apply the $t-$-interval procedure to the data? Explain your answer.

c. If you answered "yes" to part (b), find a $95\%$ confidence interval for men's average forearm length Interpret your result.

Step-by-Step Solution

Answer

Part (a) 

Part (b) The applying $t-$interval procedure to the given data is reasonable.

Part (c) The $95\%$ confidence interval for men's average forearm length is $(18.6149,18.9894)$

Part (a) Step 1: Given information

The post looked at and presented data on forearm length in inches for a group of $140$ men from the Weiss Stats website.

Part (a) Step 2: Explanation

Draw the boxplot for men's forearm length using MINITAB.

Procedure for MINITAB:

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

Step 2: Select Simple under One Y's. Click OK.

Step 3: Fill in the LENGTH data in the graph variables.

Step 4: Click the OK button.

OUTPUT FROM MINITAB: 

Draw the normal probability plot for men's forearm length 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, add the LENGTH column.

Step 4: Click the OK button.

OUTPUT FROM MINITAB: 

Part (a) Step 3: Explanation

Draw the histogram for men's forearm length using MINITAB.

Procedure for MINITAB:

Step 1: Select Histogram from the Graph menu.

Step 2: Click OK after selecting Simple.

Step 3: In Graph variables, type LENGTH in the corresponding column.

Step 4: Click the OK button. 

MINITAB output: 

Draw the stem-and-leaf diagram for men's forearm length using MINITAB.

Procedure for MINITAB:

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

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

Step 3: Choose OK.

OUTPUT FROM MINITAB: 

Stem-and-Leaf Display: LENGTH

Stem-and-leaf of LENGTH $N=140$

Leaf Unit $=0.10$

$$\begin{gathered} 4 16 1334 \\ 7 16 689 \\ 19 17 111122333344 \\ 30 17 55567778999 \\ 51 18 000111122333333334444 \\ \left(25\right) 18 555555555666677778888899 \\ 64 19 0000001111122233334444 \\ 41 19 5555666667778889999 \\ 22 20 0001123444 \\ 12 20 5555667899 \\ 2 21 04 \end{gathered}$$

Part (b) Step 1: Explanation

Check whether using the $t-$interval technique on the given data is appropriate.

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

Small Size of sample:

• From the population, samples are drawn at random.
• The sample size is higher or the population follows a normal distribution.
• The standard deviation has not been determined.

Explanation:

It is evident from section (a) that the data does not contain any outliers. The sample size is also $(=140)$ greater. As a result, men's forearm length distribution is quite symmetric. As a result, it is fair to apply the $t-$interval technique on the given data.

Part (c) Step 1: Explanation

Compute the $95\%$ confidence interval for men's average forearm length by using the MINITAB.

Procedure for MINITAB:

First, select Stat > Basic Statistics > $1-Sample$$\mathit{t}$ example

Step 2: Select SPEED from the Samples in Column menu.

Step 3: Select Options and set the Confidence Level to $95$

Step 4: In the alternative, select not equal.

Step 5: In all dialogue boxes, click OK.

OUTPUT FROM MINITAB: One-Sample T: LENGTH

Variable	N	Mean	StDev	SE Mean	95% CI
Length	140	18.8021	1.1205	0.0947	(18.6149, 18.9894)

The $95\%$ confidence interval for the men's average forearm length is $(18.6149,18.9894)$, according to the MINITAB output.