Question

An auto dealer would like to determine if there is a difference in the braking distance (the number of feet required to go from 60mph

to 0mph
) 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 60mph
. 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 156
155
155
154
153
147
Model B 161
157
159
154
153
148

Step 2 of 3 : Compute the value of the test statistic. Round your answer to three decimal places.

Answer 1

An auto dealer's t-test failed to reject the null hypothesis, indicating no significant difference in braking distances between two sedan models.

Here is the computation of the value of the test statistic:

Step 1: Formulate the null and alternative hypotheses

Null hypothesis (H₀): There is no significant difference in the braking distances of the two models.

Alternative hypothesis (H₁): There is a significant difference in the braking distances of the two models.

Step 2: Calculate the sample means and pooled variances

Mean(Model A) = 153.5

Mean(Model B) = 156

Variance(Model A) = 5.25

Variance(Model B) = 6.25

Step 3: Compute the pooled variance

Pooled Variance = (Σ(nᵢ - 1)sᵢ²) / (Σnᵢ - 2)

= (5*(5.25) + 5*(6.25)) / (6 - 2)

= 5.75

Step 4: Calculate the standard error of the mean difference

SE(Mean Difference) = √[(Pooled Variance) / (n₁ + n₂)]

= √[(5.75) / (6 + 6)]

= 1.30

Step 5: Calculate the test statistic

t = (Mean(Model B) - Mean(Model A)) / SE(Mean Difference)

= (156 - 153.5) / 1.30

= 1.92

Step 6: Determine the p-value

The p-value is the probability of obtaining a test statistic as extreme or more extreme than the one calculated, assuming the null hypothesis is true. In this case, we are using a two-tailed test with α=0.01. The p-value for a two-tailed test is calculated by doubling the one-tailed p-value.

p-value = 2 * P(t > 1.92)

Using a t-distribution table or calculator, we can find that the p-value is approximately 0.055.

Step 7: Make a decision about the null hypothesis

Since the p-value (0.055) is greater than the significance level (0.01), we fail to reject the null hypothesis. In other words, there is not enough evidence to conclude that there is a significant difference in the braking distances of the two models.

Conclusion

The auto dealer cannot conclude that there is a significant difference in the braking distances of the two models of high-end sedans.