Questions: Blocking refers to the idea that the variability in a variable can be reduced by segmenting the data by some other variable. The data in the accompanying table represent the recumbent length (in centimeters) of a sample of 10 males and 10 females who are 40 months of age. Complete parts (a) through (d). Click the icon to view the data table. (a) Determine the standard deviation of recumbent length for all 20 observations. cm (Round to two decimal places as needed.) (b) Determine the standard deviation of recumbent length for the males. cm (Round to two decimal places as needed.) (c) Determine the standard deviation of recumbent length for the females. cm (Round to two decimal places as needed.) (d) What effect does blocking by gender have on the standard deviation of recumbent length for each gender?

Solution Steps

To solve the given questions, we will follow these steps:

(a) Calculate the standard deviation for all 20 observations by combining the data for both males and females.

(b) Calculate the standard deviation for the recumbent length of the males only.

(c) Calculate the standard deviation for the recumbent length of the females only.

(d) Compare the standard deviations calculated in parts (b) and (c) to the one in part (a) to understand the effect of blocking by gender.

To solve the given problem, we will calculate the standard deviation for the entire dataset, as well as separately for males and females. We will follow the steps outlined below:

First, we need to calculate the mean of the recumbent lengths for all 20 observations. Let's denote the recumbent lengths as \( x_1, x_2, \ldots, x_{20} \).

\[ \bar{x} = \frac{1}{20} \sum_{i=1}^{20} x_i \]

The standard deviation is calculated using the formula:

\[ s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2} \]

where \( n = 20 \).

Calculate the mean of the recumbent lengths for the 10 males.

\[ \bar{x}_{\text{males}} = \frac{1}{10} \sum_{i=1}^{10} x_i \]

Use the standard deviation formula for the male observations:

\[ s_{\text{males}} = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x}_{\text{males}})^2} \]

where \( n = 10 \).

Calculate the mean of the recumbent lengths for the 10 females.

\[ \bar{x}_{\text{females}} = \frac{1}{10} \sum_{i=1}^{10} x_i \]

Use the standard deviation formula for the female observations:

\[ s_{\text{females}} = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x}_{\text{females}})^2} \]

where \( n = 10 \).

Final Answer