A small machine shop makes a part for Harley Davidson Motorcycles that needs to be 18.0 mm in diameter. The team that produces the part does a daily quality control check by randomly selecting n = 12 parts (using a systematic random sampling method). One day of operation the SRS produced a sample mean diameter of 17.96. Historically the machinery making the part has a population standard deviation of 0.09mm. We are investigating to see of the sample mean is significant at the 5% level.

What is the test statistic for this test of statistical significance?

Question

A small machine shop makes a part for Harley Davidson Motorcycles that needs to be 18.0 mm in diameter. The team that produces the part does a daily quality control check by randomly selecting n = 12 parts (using a systematic random sampling method). One day of operation the SRS produced a sample mean diameter of 17.96. Historically the machinery making the part has a population standard deviation of 0.09mm. We are investigating to see of the sample mean is significant at the 5% level.

What is the test statistic for this test of statistical significance?

BlackTom AI · Accepted Answer

First, let’s restate the problem in our own words to ensure understanding: we are testing whether the sample mean diameter of 17.96 mm, based on n = 12 parts, provides evidence that the true mean differs from the target 18.0 mm, using a 5% significance level and a known population standard deviation of 0.09 mm. The appropriate test statistic here is a z statistic because the population standard deviation is given.

Compute the standard error of the mean: SE = sigma / sqrt(n) = 0.09 / sqrt(12). Since sqrt(12) ≈ 3.464, SE ≈ 0.09 / 3.464 ≈ 0.02598 mm.

Next, calculate the z-statistic: z = (xbar - mu0) / SE = (17.96 - 18.00) / 0.02598 = (-0.04) / 0.02598 ≈ -1.539. Rounding to two decimals gives approximately -1.54.

Now, evaluate the result against the 5% significance level for a two-sided test. The critical z-values are approximately ±1.96. Our computed z ≈ -1.54 lies within (-1.96, 1.96), so it does not fall in the rejection region.

Option analysis:
- If the option is -1.54: This value matches the calculated z-statistic, so it is the correct test statistic for this scenario. The negative sign indicates the sample mean is below the hypothesized mean of 18.0 mm, which is consistent with the data (17.96 mm).

Note on potential other options: Since the problem only provides a single numerical choice in the provided data, there are no alternative test-statistic values to evaluate here. Any other value would not reflect the computation above, and would misrepresent either the mean difference or the standard error.

In summary, the computed z-statistic is approximately -1.54, which does not exceed the critical threshold of ±1.96, so at the 5% level there is not enough evidence to conclude the mean diameter differs from 18.0 mm.

STAT 130-007 Chapter 17 Quiz

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