ATUW Motors is planning to replace the brake pad of its cars with a similar quality brake pads but cheaper. This is part of the company strategy to cut costs but wants to maintain the quality of its cars and parts. The variance of the existing brake pads life time is 29.2 months. A random sample of 24 cars gave a sample variance of 27.6 months. Determine if the new brake pads performs as better as the existing brake pads
To determine if the new brake pads perform as well as the existing brake pads, we can conduct a hypothesis test using the F-test for equality of variances. The null hypothesis (H0) is that the variances of the two populations are equal, while the alternative hypothesis (H1) is that the variance of the new brake pads is significantly different from the existing brake pads.
Here are the steps to conduct the hypothesis test:
Step 1: State the hypotheses.
- Null hypothesis (H0): Var(new brake pads) = Var(existing brake pads)
- Alternative hypothesis (H1): Var(new brake pads) ≠ Var(existing brake pads)
Step 2: Select a significance level.
You need to determine the desired level of significance (α) for the test. Common choices are 0.05 or 0.01. Let's assume α = 0.05.
Step 3: Calculate the test statistic.
The test statistic for comparing variances is the ratio of the sample variances, which follows an F-distribution. The formula is: F = (sample variance of new brake pads) / (sample variance of existing brake pads)
In this case, the sample variance of the new brake pads is 27.6 months, and the sample variance of the existing brake pads is 29.2 months. Plugging these values into the formula, we get F = 27.6 / 29.2.
Step 4: Determine the critical value.
To determine if the test statistic is statistically significant, we compare it to the critical value from the F-distribution table. The critical value depends on the desired significance level (α) and the degrees of freedom for the numerator (new brake pads) and denominator (existing brake pads) samples.
The degrees of freedom for the numerator is the sample size of the new brake pads minus 1 (24 - 1 = 23), and the degrees of freedom for the denominator is the sample size of the existing brake pads minus 1 (unknown in this case).
Step 5: Make a decision.
If the test statistic is greater than the critical value or falls in the critical region, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Step 6: Interpret the decision.
If we reject the null hypothesis, it means that there is sufficient evidence to suggest that the variance of the new brake pads is different from the existing brake pads. If we fail to reject the null hypothesis, it indicates that there is not enough evidence to conclude that the variance of the new brake pads is different.
Note: In this case, since we don't have the sample size of the existing brake pads, we cannot determine the exact critical value or make a final decision. To further analyze this, you would need to collect additional data or provide the sample size of the existing brake pads.