Question

day 3 chapter 7: 7.32 b & d practice
day 3: practice problem 7.32b classwork
are jet skis dangerous? propelled by a stream of pressurized water, jet skis and other so - called wet bikes
carry from one to three people, retail for an average price of $5700, and have become one of the most
popular types of recreational vehicle sold today. but critics say that theyre noisy, dangerous, and
damaging to the environment. an article in the august 1997 issue of the journal of the american
medical association reported on a survey that tracked emergency room visits at randomly selected
hospitals nationwide. the study recorded data on the number of jet skis in use and the number of
accidents for the years 1987 - 1996. computer output and a residual plot from a linear regression
alysis of the data are shown below.
predictor coef se coef t p
constant - 0.8 109.9 - 0.01 0.994
jetskis 0.0048308 0.0002292 21.08 0.000
s = 188.3 r - sq = 98.2% r - sq(adj) = 98.0%
residuals versus jetskis

1. what is the equation of the least - squares line? be sure to define any variables you use.
2. interpret the value of ( r^{2} ) in the context of this problem.
3. is a line an appropriate model for these data? justify your answer.
4. interpret the value of s in the context of this problem.

Explanation:

1.

Step1: Identify variables

Let \( y \) be the number of accidents and \( x \) be the number of jet - skis in use.

Step2: Use regression equation formula

The general form of a linear regression equation is \( \hat{y}=b_0 + b_1x \), where \( b_0 \) is the constant (intercept) and \( b_1 \) is the coefficient for the predictor variable.
From the table, \( b_0=- 0.8 \) and \( b_1 = 0.0048308 \)
So the equation is \( \hat{y}=-0.8+0.0048308x \)

\( R - Sq=98.2\%\). The coefficient of determination \( r^{2}\) (here \( R - Sq\) is equivalent to \( r^{2}\)) represents the proportion of the variance in the response variable (number of accidents) that is predictable from the predictor variable (number of jet - skis). So, 98.2% of the variation in the number of accidents can be explained by the linear relationship with the number of jet - skis in use.

Looking at the residual plot:

• If the residual plot shows a random scatter of points around the horizontal axis (residual = 0), a line is an appropriate model.
• In this residual plot, although there are a few points that deviate, there is no clear pattern (such as a curve, increasing or decreasing trend). Also, \( r^{2}=98.2\%\) is very high, indicating a strong linear relationship.

Answer:

The equation of the least - squares line is \( \hat{y}=-0.8 + 0.0048308x \), where \( \hat{y} \) is the predicted number of accidents and \( x \) is the number of jet - skis in use.

2.