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QuestionAnna Bischoff 12/03/24 10:44 AM Question 13, 9.1.33-T Part 2 of 4 HW Score: 66.33%,9.2966.33 \%, 9.29 of 14 points Points: 0 of 1 Save
The weights (in pounds) of eight vehicles and the variabilities of their braking distances (in feet) when stopping on a dry surface are shown in the table. At α=0.05\alpha=0.05, is there enough evidence to conclude that there is a significant linear correlation between vehicle weight and variability in braking distance on a dry surface? \begin{tabular}{|l|c|c|c|c|c|c|c|c|} \hline Weight, x\mathbf{x} & 5910 & 5360 & 6500 & 5100 & 5890 & 4800 & 5700 & 5870 \\ \hline Variability, y\mathbf{y} & 1.72 & 1.99 & 1.92 & 1.55 & 1.69 & 1.50 & 1.57 & 1.70 \\ \hline \end{tabular}
Setup the hypothesis for the test. H0:ρ=0Ha:ρ≠0\begin{array}{l} H_{0}: \rho=0 \\ H_{a}: \rho \neq 0 \end{array}
Calculate the test statistic. t=3.33\mathrm{t}=3.33 (Round to two decimal places as needed.)

Studdy Solution

STEP 1

1. We are testing for a linear correlation between vehicle weight and variability in braking distance.
2. The significance level is α=0.05\alpha = 0.05.
3. The null hypothesis H0H_0 is that there is no linear correlation (ρ=0\rho = 0).
4. The alternative hypothesis HaH_a is that there is a linear correlation (ρ≠0\rho \neq 0).

STEP 2

1. Setup the hypothesis for the test.
2. Calculate the test statistic.
3. Determine the critical value and compare it with the test statistic.
4. Make a decision based on the comparison.

STEP 3

Setup the hypothesis for the test:
H0:ρ=0Ha:ρ≠0\begin{array}{l} H_{0}: \rho=0 \\ H_{a}: \rho \neq 0 \end{array}

STEP 4

Calculate the test statistic. The test statistic t t is given as:
t=3.33t = 3.33

STEP 5

Determine the critical value for a two-tailed test at α=0.05\alpha = 0.05 with n−2=8−2=6n - 2 = 8 - 2 = 6 degrees of freedom. Use a t-distribution table or calculator to find the critical value.

STEP 6

The critical value for α=0.05 \alpha = 0.05 and 6 degrees of freedom is approximately ±2.447 \pm 2.447 .

STEP 7

Compare the test statistic with the critical values. Since ∣3.33∣>2.447 |3.33| > 2.447 , we reject the null hypothesis.
Conclusion: There is enough evidence to conclude that there is a significant linear correlation between vehicle weight and variability in braking distance on a dry surface.

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