In a study of distances travelled by buses before the first major engine failure, a sampling of 60 buses resulted in a mean of 152,600 km. This means that our sample had their first major engine failure at an average of 152,600 km. At the 0.05 level of significance, we are tasked with testing the manufacturer's claim that the mean distance traveled before a major engine failure is more than 154,000 km. From previous research, the population standard deviation is known to be 6,400 km.

Step 1: State hypotheses

• Null hypothesis (H₀): μ ≤ 154,000 km (the mean distance is less than or equal to 154,000 km)
• Alternative hypothesis (H₁): μ > 154,000 km (the mean distance is more than 154,000 km)

Step 2: Set the significance level

The significance level (α) is 0.05, indicating a 5% risk of rejecting the null hypothesis when it is true.

Step 3: State the decision rule

Using the standard normal distribution (z-test), we find the critical value zₐ for a one-tailed test at α=0.05, which is approximately 1.645. The decision rule is: if the calculated z value exceeds 1.645, reject H₀; otherwise, do not reject H₀.

Step 4: Calculate the test statistic

The formula for the z-test is:

z = (x̄ - μ₀) / (σ / √n)

where:

• x̄ = 152,600 km (sample mean)
• μ₀ = 154,000 km (claimed population mean)
• σ = 6,400 km (population standard deviation)
• n = 60 (sample size)

Calculating:

z = (152,600 - 154,000) / (6,400 / √60) = (-1,400) / (6,400 / 7.746) ≈ (-1,400) / 825.589 ≈ -1.696

Step 5: State the conclusion

Since the calculated z-value is approximately -1.696, which is less than 1.645, we fail to reject the null hypothesis at the 0.05 significance level. Therefore, there is not enough evidence to support the manufacturer's claim that the mean distance before engine failure exceeds 154,000 km.

Complementary Discussion on the Study

The results of this hypothesis test clarify the reliability of the buses in terms of engine longevity. The observed sample mean of 152,600 km is slightly below the claimed average of 154,000 km, and the test indicates that this difference is not statistically significant at the 5% level. Hence, the data do not provide sufficient evidence to reject the null hypothesis, implying that the manufacturer’s claim cannot be statistically validated based on this sample.

A Study Of Distances Travelled By Buses Before

The process highlights the importance of hypothesis testing in quality control and product reliability studies. In practical applications, such findings inform manufacturers and stakeholders about the actual performance versus claimed standards, which can affect maintenance schedules, warranty policies, and consumer confidence. It also exemplifies the application of z-tests when the population standard deviation is known, a common scenario in industrial quality assessments.

Additional Considerations and Assumptions

It is important to recognize assumptions underpinning the z-test used here: the sample of buses is randomly selected, the population standard deviation is accurate, and the distribution of the distances is approximately normal. If any of these assumptions are violated, alternative tests such as the t-test would be appropriate, especially for smaller samples or unknown population variances. Furthermore, the conclusion hinges on the significance level; at a different α, the decision might change, influencing interpretations of reliability data.

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