Question

Exercises 13–28 use the same data sets as Exercises 13–28 in Section 10-1. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted value by following the prediction procedure summarized in Figure 10-5 on page 493.

Use the foot lengths and heights to find the best predicted height of a male

who has a foot length of 28 cm. Would the result be helpful to police crime scene investigators in trying to describe the male?

Short answer

The regression equation is\(\hat y = 125 + 1.73x\).

The best-predicted value is the mean height of 177 cm. It will not be helpful to the police in trying to obtain a description of the male.

Step-by-step solution

Given information

The given data provides the information of the shoe print (in cm) and the height (in cm), as follows.

State the equation for the regression linea

The formula for the estimated regression line is

\(y = {b_0} + {b_1}x\).

Here,

\({b_0}\)is the Y-intercept,

\({b_1}\)is the slope,

\(x\)is the explanatory variable, and

\(\hat y\)is the response variable (predicted value).

Let X denote the foot length (in cm) and Y denote the height (in cm) of the male.

Compute the slope and intercept

The calculations required to compute the slope and intercept are as follows.

The sample size is \(\left( n \right) = 5\).

The slope is computed as

\(\begin{array}{c}{b_1} = \frac{{n\left( {\sum {xy} } \right) - \left( {\sum x } \right)\left( {\sum y } \right)}}{{n\left( {\sum {{x^2}} } \right) - {{\left( {\sum x } \right)}^2}}}\\ = \frac{{5 \times 23209.27 - 130.8 \times 886.5}}{{5 \times 3426.96 - {{130.8}^2}}}\\ = 3.5226\end{array}\).

The intercept is computed as

\(\begin{array}{c}{b_0} = \frac{{\left( {\sum y } \right)\left( {\sum {{x^2}} } \right) - \left( {\sum x } \right)\left( {\sum {xy} } \right)}}{{n\left( {\sum {{x^2}} } \right) - {{\left( {\sum x } \right)}^2}}}\\ = \frac{{886.5 \times 3426.96 - 130.8 \times 23209.27}}{{5 \times 3426.96 - {{130.8}^2}}}\\ = 85.15\end{array}\).

The estimated regression equation is

\(\begin{array}{c}\hat y = {b_0} + {b_1}x\\ = 85.15 + 3.5226x\end{array}\).

Checking the model

Refer to exercise 18 of section 10-1 for the following result.

1) The scatter plot does not show an approximate linear relationship between the variables.

2) The P-value is 0.085.

As the P-value is greater than the level of significance (0.05), the null hypothesis is failed to be rejected.

Therefore, the correlation is not significant.

Referring to figure 10-5, the criteria for a good regression model are not satisfied.

The best-predicted value of a variable is simply its sample mean.

Compute the prediction 

The best-predicted height of a male who has a foot length of 28 cm is obtained as the mean of the sample responses.

The sample mean is computed as

\(\begin{array}{c}\bar y = \frac{{\sum\limits_{i = }^n {{y_i}} }}{n}\\ = \frac{{\left( {175.3 + 177.8 + ... + 172.7} \right)}}{5}\\ = 177.3\end{array}\).

Therefore, the best-predicted height of the male who has a foot length of 28 cm will be 177 cm. As the best-predicted value is the mean height (177 cm), it will not help the police to describe the male.