Chapter 27, Problem 42E

a)

To determine

To check the assumptions and conditions for inference.

Answer to Problem 42E

All conditions satisfied.

Explanation of Solution

Given:

  

The conditions are: straight enough, Independence, Randomization, Random residual, Does plot thicken? and Nearly normal condition.

We will check it one by one.

Straight enough condition: Satisfied, because no curvature is present in the scatterplot.

Independence assumption: Satisfied, because the 18 homes are less than 10% of the population of all homes.

Randomization condition: Satisfied, as homes were randomly selected.

Random residuals condition: Satisfied, because there is no obvious pattern in the residual plot.

Does the plot thicken? condition: Satisfied, because the vertical spread in the residual plot is not increasing or decreasing.

b)

To determine

To perform the hypothesis test.

Answer to Problem 42E

There is sufficient evidence that the slope is non zero and there is association between size and the assessed value.

Explanation of Solution

Given:

  

Formula:

Test statistic:

  $t=\frac{b-{\beta }_0}{S{E}_b}$

The null and alternative hypotheses:

  $\begin{array}{l}{H}_0:{\beta }_1=0\\ H_a:{\beta }_1\ne 0\end{array}$

Test statistic:

  $t=\frac{11.8987-0}{4.290}=2.774$

The degrees of freedom = df = 16

Therefore, p-value would be,

P-value = 0.0135 …Using excel formula, =TDIST(2.774,16,2)

Decision: P-value < 0.05, reject H₀.

Conclusion: There is sufficient evidence that the slope is non zero and there is association between size and the assessed value.

c)

To determine

To explain the percentage of the variability in assessed value is explained by this regression.

Answer to Problem 42E

32.5% of the variation in the assessed values has been explained by model.

Explanation of Solution

Given:

  

The R-square = 32.5%.

Hence, the 32.5% of the variation in the assessed values has been explained by model.

d)

To determine

To find the 90% confidence interval for the slope of the regression line.

Answer to Problem 42E

We are 90% confident that the assessed value increases between 4.4085 and 19.3890.

Explanation of Solution

Given:

  

Formula:

Confidence interval for slope is,

  $(Coefficient(b)\mp t_c\times S{E}_b)$

The confidence level = 0.90

So, level of significance = $\alpha$ = 0.10

First need to find critical t value.

tc = 1.746 …Using excel formula, =TINV(0.10,16)

Using formula, 90% confidence interval for the slope of the regression line.

  $\begin{array}{l}(11.8987-1.746\times 4.290,11.8987+1.746\times 4.290)\\ (4.4085,19.3890)\end{array}$

Therefore, we are 90% confident that the assessed value increases between 4.4085 and 19.3890.

e)

To determine

To explain whether we conclude that adding a room to your house will increase its assessed value.

Answer to Problem 42E

No.

Explanation of Solution

Given:

  

The association does not mean causation. Therefore, adding a room does not necessarily cause an increase in the assessed value. There might be possibility that the confounding variable affect it.

f)

To determine

To explain whether assessed value $70,200 is too high.

Answer to Problem 42E

Yes, the assessed value $70,200 is too high.

Explanation of Solution

Given:

  

The regression equation is,

  $\widehat{y}=37108.8+11.8987\times x$

The value $x$ = 2100.

  $\widehat{y}=37108.8+11.8987\times 2100=62096.07$

Therefore, $70,200 is higher than $62096.07. Hence, it is too high.