Chapter 12.1, Problem 13E

(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 the weeds among the corn yields. 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 seem to roughly lie about a line.

Strength: Weak, because the points lie far apart.

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

(b)

To determine

To find out what is the equation of the least square regression line for predicting corn yield from the number of lamb’s quarter plants per meter.

Answer to Problem 13E

  $\begin{array}{l}\widehat{y}=a+bx\\ \Rightarrow \widehat{y}=166.483-1.0987x\end{array}$

Explanation of Solution

Now, as we know that in the question, the researcher examined the weeds among the corn yields 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”:

  $\begin{array}{l}\widehat{y}=a+bx\\ \Rightarrow \widehat{y}=166.483-1.0987x\end{array}$

with $\widehat{y}$ the predicted corn yield and x the number of weeds per meter.

(c)

To determine

To interpret the slope and y -intercept of the regression line in this context.

Explanation of Solution

Now, as we know that in the question, the researcher examined the weeds among the corn yields 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”:

  $\begin{array}{l}\widehat{y}=a+bx\\ \Rightarrow \widehat{y}=166.483-1.0987x\end{array}$

with $\widehat{y}$ the predicted corn yield and x the number of weeds per meter.

Thus, the slope b is the coefficient of x and is thus $-1.0987$ . This means that the corn yield is expected to decrease by $1.0987$ bushels per weed per meter.

And the y -intercept a is the constant of the regression equation and in thus $166.483$ . This means that the cor yield is expected to be $166.483$ when there are zero weeds per meter.

(d)

To determine

To explain do these data provide convincing evidence that more weeds reduce corn yield and carry out an appropriate test at the $\alpha =0.05$ level.

Answer to Problem 13E

Yes, there is sufficient evidence to support the claim that more weeds reduce corn yield.

Explanation of Solution

It is given in the computer output that:

  $\begin{array}{l}n=16\\ b=-1.0987\\ {\text{SE}}_b=0.5712\end{array}$

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:

  $\begin{array}{l}t=\frac{b-\beta }{{\text{SE}}_b}\\ =\frac{-1.0987-0}{0.5712}\\ =-1.923\end{array}$

And the degrees of freedom is:

  $\begin{array}{l}df=n-2\\ =16-2\\ =14\end{array}$

Thus, the P-value is as:

  $0.025<P<0.05$

Or using technology the P-value is as:

  $P=0.03753$

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

  $P<0.05\Rightarrow \text{ Reject }{H}_0$

Thus, we conclude that there is sufficient evidence to support the claim that more weeds reduce corn yield.