Chapter 12.1, Problem 14E

(a)

To determine

To describe what this graph tells you about the relationship between these two variables.

Explanation of Solution

In the question the researcher examined does how long young children remain at the lunch table help predict how much they eat. Thus, the scatterplot is also given in the question for the variables used. Thus, by looking at the scatterplot we can say that,

Direction: Negative, because the scatterplot slopes downwards

Form: Linear, because the points seems to roughly lie about a line.

Strength: Moderate, because the points do not lie far apart, but not close together either.

Thus, the scatterplot suggest a moderate negative linear relationship between the variables.

(b)

To determine

To find out what is the equation of the least square regression line for predicting calories consumed from the time at the table.

Answer to Problem 14E

  $\begin{array}{l}\widehat{y}=a+bx\\ \Rightarrow \widehat{y}=560.65-3.0771x\end{array}$

Explanation of Solution

Now, in the question the researcher examined does how long young children remain at the lunch table help predict how much they eat. and the computer output of this data is given. And also as we know the general regression equation be as:

  $\widehat{y}=a+bx$

And the estimate a and b are given in the column of the computer output “Coef”:

  $\begin{array}{l}\widehat{y}=a+bx\\ \Rightarrow \widehat{y}=560.65-3.0771x\end{array}$

with $\widehat{y}$ the predicted calories and x the time at the table.

(c)

To determine

To interpret the slope of the regression line in this context and explain does it make sense to interpret the y -intercept in this case, why or why not.

Answer to Problem 14E

No, it does not make sense to interpret the y -intercept in this case.

Explanation of Solution

Now, as we know that in the question the researcher examined does how long young children remain at the lunch table help predict how much they eat. and the computer output of this data is given. And also as we know the general regression equation be as:

  $\widehat{y}=a+bx$

And the estimate a and b are given in the column of the computer output “Coef”:

  $\begin{array}{l}\widehat{y}=a+bx\\ \Rightarrow \widehat{y}=560.65-3.0771x\end{array}$

with $\widehat{y}$ the predicted calories and x the time at the table.

Thus, the slope b is the coefficient of x and is thus $-3.0771$ . This means that the calories are expected to decrease by $3.0771$ calories per minute.

And the y -intercept a is the constant of the regression equation and in thus $560.65$ . This means that the calories are expected to be $560.65$ after zero minute. Thus, it does not make sense to interpret y -intercept as in actual it does not be zero.

(d)

To determine

To explain do these data provide convincing evidence of a negative linear relationship between time at the table and calories consumed in the population of toddlers and carry out an appropriate test at the $\alpha =0.01$ level.

Answer to Problem 14E

Yes, there is sufficient evidence to support the claim of a negative linear relationship between time at the table and calories consumed in the population of toddlers.

Explanation of Solution

It is given in the computer output that:

  $\begin{array}{l}n=20\\ b=-3.0771\\ {\text{SE}}_b=0.8498\end{array}$

Thus, we define the hypothesis by:

  $\begin{array}{l}{H}_0:\beta =0\\ H_a:\beta <0\end{array}$

Thus, the value of the test statistics is as:

  $\begin{array}{l}t=\frac{b-\beta }{{\text{SE}}_b}\\ =\frac{-3.0771-0}{0.8498}\\ =-3.621\end{array}$

And the degrees of freedom is:

  $\begin{array}{l}df=n-2\\ =20-2\\ =18\end{array}$

Thus, the P-value is as:

  $0.005<P<0.01$

As we know that if the P-value is less than or equal to the significance level, then the null hypothesis is rejected as:

  $P<0.01\Rightarrow \text{ Reject }{H}_0$

Thus, we conclude that there is sufficient evidence to support the claim of a negative linear relationship between time at the table and calories consumed in the population of toddlers.