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)
Try an independent groups t-test.
Hypotheses:
Ho: µ1 = µ2 -->population means are equal
Ha: µ1 does not equal µ2 -->population means are not equal
Use (n1 + n2 - 2) degrees of freedom for this test. Use a t-table to determine your cutoff or critical value to reject the null using 0.01 level of significance for a two-tailed test. If your test statistic exceeds the critical values from the table, reject the null and conclude a difference (µ1 does not equal µ2). If the test statistic does not exceed the critical values from the table, do not reject the null.
I hope this will help get you started.
To determine whether we can reject the insurance claim made by Metlife Insurance, we need to conduct a hypothesis test. In this case, we will compare the means of two independent samples (Toyota Corolla and Toyota Rav4) to see if they are significantly different.
Let's define our hypotheses:
- Null Hypothesis (H0): The mean footwell intrusions for Toyota Corolla and Toyota Rav4 are equal.
- Alternative Hypothesis (H1): The mean footwell intrusions for Toyota Corolla and Toyota Rav4 are not equal.
Next, we need to determine the significance level (α) for the hypothesis test. In this case, it is given as α = 0.01, which means we need to be 99% confident in our result.
Since we don't know the population variances in this case, we will use the t-test for independent samples.
Now, let's calculate the test statistic and compare it to the critical value to make our decision:
1. Calculate the pooled standard deviation (Sp) using the formula:
Sp = sqrt(((n1 - 1) * s1^2 + (n2 - 1) * s2^2) / (n1 + n2 - 2))
where n1 and n2 are the sample sizes, and s1 and s2 are the sample standard deviations.
Sp = sqrt(((7 - 1) * 4.53^2 + (13 - 1) * 3.84^2) / (7 + 13 - 2))
Sp = sqrt((6 * 20.5209 + 12 * 14.7456) / 18)
Sp = sqrt((123.1254 + 176.9472) / 18)
Sp = sqrt(300.0726 / 18)
Sp = sqrt(16.6707)
Sp = 4.0863 (rounded to 4 decimal places)
2. Calculate the test statistic (t):
t = (x1 - x2) / (Sp * sqrt(1/n1 + 1/n2))
where x1 and x2 are the sample means, and n1 and n2 are the sample sizes.
t = (11.8 - 9.52) / (4.0863 * sqrt(1/7 + 1/13))
t = 2.28 / (4.0863 * sqrt(0.1429 + 0.0769))
t = 2.28 / (4.0863 * sqrt(0.2198))
t = 2.28 / (4.0863 * 0.4690)
t = 2.28 / 2.0300
t = 1.1239 (rounded to 4 decimal places)
3. Determine the degrees of freedom (df) using the formula: df = n1 + n2 - 2.
df = 7 + 13 - 2
df = 18
4. Look up the critical t-value based on the significance level (α = 0.01) and the degrees of freedom (df = 18). Using a t-table or a t-distribution calculator, the critical t-value is approximately ±2.878.
5. Compare the calculated t-value (1.1239) with the critical t-value (±2.878).
Since the calculated t-value (1.1239) is less than the critical t-value (±2.878), we fail to reject the null hypothesis. This means there is not enough evidence to support the claim that the mean footwell intrusions for Toyota Corolla and Toyota Rav4 are different at a significance level of 0.01.
Therefore, we cannot reject the insurance claim made by Metlife Insurance.