Question

question 2 of 6, step 2 of 3
an auto dealer would like to determine if there is a difference in the braking distance (the number of feet required to go from 60 mph to 0 mph) of two different models of a high - end sedan. six drivers are randomly selected and asked to drive both models and brake once they have reached exactly 60 mph. the distance required to come to a complete halt is then measured in feet. the results of the test are as follows. can the auto dealer conclude that there is a significant difference in the braking distances of the two models? use α = 0.01. let the braking distances of model a represent population 1 and the braking distances of model b represent population 2.
braking distance of high - end sedans (feet)
driver 1 2 3 4 5 6
model a 148 147 145 146 158 157
model b 150 152 147 146 159 159
copy data
step 2 of 3: compute the value of the test statistic. round your answer to three decimal places.
answer

Explanation:

Step1: Calculate the differences

Let \(d_i=x_{iA}-x_{iB}\), where \(x_{iA}\) is the braking - distance of Model A for driver \(i\) and \(x_{iB}\) is the braking - distance of Model B for driver \(i\).
\(d_1 = 148 - 150=-2\), \(d_2 = 147 - 152=-5\), \(d_3 = 145 - 147=-2\), \(d_4 = 146 - 146 = 0\), \(d_5 = 158 - 159=-1\), \(d_6 = 157 - 159=-2\).

Step2: Calculate the mean of the differences

\(\bar{d}=\frac{\sum_{i = 1}^{n}d_i}{n}\), where \(n = 6\) and \(\sum_{i=1}^{6}d_i=-2-5 - 2+0 - 1-2=-12\).
\(\bar{d}=\frac{-12}{6}=-2\).

Step3: Calculate the standard deviation of the differences

First, calculate \((d_i-\bar{d})^2\) for each \(i\):
\((d_1-\bar{d})^2=(-2 + 2)^2 = 0\), \((d_2-\bar{d})^2=(-5 + 2)^2 = 9\), \((d_3-\bar{d})^2=(-2 + 2)^2 = 0\), \((d_4-\bar{d})^2=(0 + 2)^2 = 4\), \((d_5-\bar{d})^2=(-1 + 2)^2 = 1\), \((d_6-\bar{d})^2=(-2 + 2)^2 = 0\).
\(\sum_{i = 1}^{n}(d_i-\bar{d})^2=0 + 9+0 + 4+1+0 = 14\).
The standard deviation \(s_d=\sqrt{\frac{\sum_{i = 1}^{n}(d_i-\bar{d})^2}{n - 1}}=\sqrt{\frac{14}{5}}\approx1.673\).

Step4: Calculate the test statistic

The test statistic for a paired - t test is \(t=\frac{\bar{d}-\mu_d}{s_d/\sqrt{n}}\), where \(\mu_d = 0\) (under the null hypothesis \(H_0:\mu_d = 0\)).
\(t=\frac{-2-0}{1.673/\sqrt{6}}\approx\frac{-2}{0.683}\approx - 2.928\).

Answer:

\(-2.928\)