Through extensive research, Metlife Insurance claims that the mean footwell intrusions for Toyota Corrola and Toyota Rav4 are equal. Crash test at 40mph were performed at 7 randomly selected Toyota Corrola and 13 randomly Toyota Rav4s selected. The amount that the footwell intruded on the driver's side was measured. The mean footwell intrusion for Toyota Corrola was 11.8 centimeters with a standard deviation of 4.53. The mean footwell intrusion for Toyota Rav4was 9.52 centimeters with a standard deviation of 3.84. At a=0.01 can you reject the insurance claim? Assume the population variances are samples. (Independent Samples)
(Can you provide complete detailed answer)
To determine if we can reject the insurance claim, we will perform a hypothesis test.
Step 1: State the null and alternative hypothesis.
- Null hypothesis (H0): The mean footwell intrusions for Toyota Corolla and Toyota Rav4 are equal.
- Alternative hypothesis (Ha): The mean footwell intrusions for Toyota Corolla and Toyota Rav4 are not equal.
Step 2: Set the significance level (α).
- The significance level (α) is given as 0.01.
Step 3: Calculate the test statistic.
- As the population variances are unknown, we will use the t-test for independent samples.
- The formula for calculating the t-test is:
t = (x1 - x2) / sqrt((s1^2 / n1) + (s2^2 / n2))
where x1 and x2 are the sample means, s1 and s2 are the sample standard deviations, n1 and n2 are the respective sample sizes.
Given data:
- For Toyota Corolla: sample mean (x1) = 11.8 cm, sample standard deviation (s1) = 4.53 cm, sample size (n1) = 7.
- For Toyota Rav4: sample mean (x2) = 9.52 cm, sample standard deviation (s2) = 3.84 cm, sample size (n2) = 13.
Substituting the values into the formula, we can calculate the t-value.
t = (11.8 - 9.52) / sqrt((4.53^2 / 7) + (3.84^2 / 13))
≈ 2.027
Step 4: Determine the critical value(s).
- Since our alternative hypothesis is two-tailed, we need to find the critical values for a two-tailed t-test.
- With a significance level (α) of 0.01 and degrees of freedom (df) equal to n1 + n2 - 2, we can consult the t-distribution table or use statistical software.
- For a two-tailed t-test at α = 0.01 and df = 7 + 13 - 2 = 18, the critical values are approximately ±2.898.
Step 5: Make a decision.
- If the test statistic t falls within the critical region (outside the critical values), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
In this case, the calculated t-value of 2.027 falls within the range of -2.898 to 2.898. Therefore, it does not fall outside the critical region. As a result, we fail to reject the null hypothesis.
Step 6: State the conclusion.
Based on the test results, there is not enough evidence to reject the insurance claim that the mean footwell intrusions for Toyota Corolla and Toyota Rav4 are equal.