Chapter 27, Problem 33E

a)

To determine

To find how many batteries were tested.

Answer to Problem 33E

The number of batteries tested = 33.

Explanation of Solution

Given:

  

  

  

The number of batteries tested = 33.

b)

To determine

To explain whether the conditions for inference are satisfied.

Answer to Problem 33E

All conditions are 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 33 cars are less than 10% of the population of all cars.

Randomization condition: Satisfied, assuming that the sample is representative for the population.

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.

Nearly normal condition: Satisfied, because the normal quantile plot of the residual is roughly linear.

c)

To determine

To test an appropriate hypothesis.

Answer to Problem 33E

There is sufficient evidence that the slope is non zero and provide evidence that the more expensive batteries do have more cranking amps.

Explanation of Solution

Given:

  

  

  

Formula:

Test statistic:

  $t=\frac{b-{\beta }_0}{S{E}_b}$

The null and alternative hypotheses:

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

  $H_0:$ No linear relationship between Cost and Cranking Amps.

  $H_a:$ Cranking Amps increase with Cost.

Test statistic:

  $t=\frac{4.14649-0}{1.282}=3.23$

The degrees of freedom = df = 31

Therefore, p-value would be,

P-value = 0.00146 …Using excel formula, =TDIST(3.23,31,1)

Decision: P-value < 0.05, reject H₀.

Conclusion: There is sufficient evidence that the slope is non zero and provide evidence that the more expensive batteries do have more cranking amps.

d)

To determine

To explain whether the association is strong.

Answer to Problem 33E

No.

Explanation of Solution

Given:

  

  

  

The R-square value is 25.2% which is very low and s = 116. Therefore, there is no strong relationship.

e)

To determine

To write the equation of the regression line.

Answer to Problem 33E

  $CrankingAmps=384.59+4.15\times \cos t$

Explanation of Solution

Given:

  

The general regression equation is,

  $\widehat{y}=b_0+b_{1}x$

Here,

  $\begin{array}{l}{b}_0=384.59\\ b_1=4.15\end{array}$

Therefore, the equation of regression line is,

  $CrankingAmps=384.59+4.15\times \cos t$

f)

To determine

To find the 90% confidence interval for slope.

Answer to Problem 33E

The 90% confidence interval for slope is,

  $(1.97,6.32)$

Explanation of Solution

Given:

  

Formula:

Confidence interval for slope is,

  $(b-t_c\times S{E}_b,b+t_c\times S{E}_b)$

First, need to find critical value $t_c$ .

Critical value for df = 31 and confidence level = 0.90 is,

  $t_c=1.696$ …Using excel formula, =TINV(0.10,31)

Therefore, the 90% confidence interval for slope is,

  $\begin{array}{l}(4.14649-1.696\times 1.282,4.14649+1.696\times 1.282)\\ (1.97,6.32)\end{array}$

g)

To determine

To interpret the 90% confidence interval for slope.

Answer to Problem 33E

We are 90% confident that Cranking amps increases on average between 1.97 and 6.32 per dollar of battery cost increases.

Explanation of Solution

Given:

  

The 90% confidence interval for slope is,

  $(1.97,6.32)$

We are 90% confident that Cranking amps increases on average between 1.97 and 6.32 per dollar of battery cost increases.