Question

A large car company states that the brake system will function properly for 5.3 years with a population standard deviation of 1.2 years before needing maintenance. An independent research facility is concerned that the brake systems may not last as long as the company claims. They took a random sample of 36 cars made by the manufacturer and found the average to be 5.0 years. Test the hypothesis at the 1% significance level.

a. State the null and alternative hypotheses
b. Calculate the test statistic.
c. Find the p-value
d. State your decision.
e. State the conclusion in the context of the problem

Answer 1

The null and alternative hypotheses for the given question are stated, and the test statistic and p-value are calculated. The decision and conclusion are also provided.

Step-by-step explanation:

a. Null Hypothesis (H0): The mean lifespan of brake systems is 5.3 years.

Alternative Hypothesis (Ha): The mean lifespan of brake systems is less than 5.3 years.

b. To calculate the test statistic, we can use the formula:

Test Statistic = (Sample Mean - Population Mean) / (Population Standard Deviation / √(Sample Size))

Test Statistic = (5.0 - 5.3) / (1.2 / √(36)) = -1.58113883

c. To find the p-value, we can use the Z-table or a statistical calculator. The p-value represents the probability of obtaining a test statistic as extreme as the one observed, if the null hypothesis is true. In this case, the p-value is less than 0.01, which is less than the significance level of 0.05.

d. Since the p-value is less than the significance level, we reject the null hypothesis.

e. In the context of the problem, we have sufficient evidence to conclude that the mean lifespan of the brake systems is less than the claimed 5.3 years at a 1% significance level.

Answer 2

The null hypothesis that brake systems last at least 5.3 years is rejected based on a sample showing an average lifespan of 5 years. A test statistic of -1.5 leads us to conclude that the true lifespan is less than claimed by the company, with a significance level of 1%.

Step-by-step explanation:

The student question involves testing a hypothesis regarding the lifespan of a car brake system using a sample mean. To answer the question:

1. Null and alternative hypotheses: The null hypothesis (H0) claims that brake systems last at least 5.3 years on average (H0: mu ≥ 5.3), while the alternative hypothesis (H1) suggests that the average lifespan is less than 5.3 years (H1: mu < 5.3).
2. Test statistic: Using the formula for the z-test statistic, which is (sample mean - population means)/(population standard deviation/sqrt(sample size)), we find z = (5.0 - 5.3)/(1.2/sqrt(36)) = -1.5.
3. p-value: A z-score of -1.5 corresponds to a p-value that can be found using statistical tables or software.
4. Decision: Since the p-value is less than the significance level of 0.01, we reject the null hypothesis.
5. Conclusion: With a 1% significance level, we have sufficient evidence to conclude that the brake systems last shorter than the car company claims.