A purchase manager knows that the hardness of castings from any supplier is normally distributed with a mean of 20.25 and SD of 2.5. He picks up 100 samples of castings from any supplier who claims that his castings have heavier hardness and finds the mean hardness as 20.50. Test whether the claim of the supplier is tenable.

Q. A purchase manager knows that the hardness of castings from any supplier is normally distributed with a mean of 20.25 and SD of 2.5. He picks up 100 samples of castings from any supplier who claims that his castings have heavier hardness and finds the mean hardness as 20.50. Test whether the claim of the supplier is tenable.

Problem Understanding and Setup:

The purchase manager knows that the hardness of castings from any supplier is normally distributed with a mean ( $\mu$ ) of 20.25 and a standard deviation ( $\sigma$ ) of 2.5. The supplier claims that his castings have a higher average hardness, and the manager takes a sample of 100 castings. From this sample, he finds a mean hardness of 20.50. The task is to test whether this observed mean supports the supplier's claim that the hardness of his castings is greater than the historical mean of 20.25.

Step 1: Formulating the Hypotheses

In hypothesis testing, we typically start by defining the null hypothesis ( $H_0$ ) and the alternative hypothesis ( $H_1$ ):

· Null Hypothesis ( $H_0$

$H_0: \mu = 20.25$

· Alternative Hypothesis ( $H_1$

$H_1: \mu > 20.25$

Step 2: Determining the Test Statistic

Since the population standard deviation ( $\sigma = 2.5$ ) is known and we have a sample of size $n = 100$ , we can use the Z-test for the sample mean. The Z-test statistic is given by the formula:

$Z = \frac{\bar{X} - \mu_0}{\frac{\sigma}{\sqrt{n}}}$

Where:

$\bar{X} = 20.50$
$\mu_0 = 20.25$
$\sigma = 2.5$
n=100n = 100n=100 is the sample size. Now, plugging in the known values: $Z = \frac{20.50 - 20.25}{\frac{2.5}{\sqrt{100}}} = \frac{0.25}{\frac{2.5}{10}} = \frac{0.25}{0.25} = 1$
So, the calculated Z-score is 1.
Step 3: Determining the Critical Value
To determine whether the result is statistically significant, we need to compare the calculated Z-score to the critical Z-value at a chosen significance level ( $\alpha$ ).
Commonly used significance levels are $\alpha = 0.05$ (5%) or $\alpha = 0.01$ (1%). In this case, let’s assume a significance level of 0.05, which is typical in hypothesis testing.
For a one-tailed test with $\alpha = 0.05$ , we look up the critical Z-value in the standard normal distribution table. The critical Z-value corresponding to a right-tailed test at $\alpha = 0.05$ is 1.645.
Step 4: Decision Rule

If the calculated Z-score is greater than the critical Z-value, we reject the null hypothesis.

If the calculated Z-score is less than or equal to the critical Z-value, we fail to reject the null hypothesis.

In this case, the calculated Z-score is 1, which is less than the critical value of 1.645.
Step 5: Conclusion
Since the calculated Z-score (1) is less than the critical value (1.645), we fail to reject the null hypothesis. This means that there is insufficient evidence to support the supplier's claim that the hardness of his castings is significantly greater than the historical mean of 20.25.
Thus, based on the sample data, the claim of the supplier that his castings have heavier hardness than the historical average is not tenable at the 5% significance level.
Step 6: Power of the Test
While the conclusion of the hypothesis test is clear, it might also be useful to consider the power of the test — that is, the probability of correctly rejecting the null hypothesis when the alternative hypothesis is true.
In this case, the power of the test depends on factors such as the sample size, the true mean hardness of the supplier's castings, and the significance level. Since the sample size is 100, which is relatively large, and the effect size (the difference between the sample mean and population mean) is small (only 0.25), the power of the test may not be very high in this specific case. If the supplier’s castings truly have a significantly higher hardness, a larger sample size or a more substantial difference between the sample mean and population mean would improve the test's power.
Step 7: Reconsidering the Analysis
If the supplier insists on the claim and wants to provide more evidence, there are a few things that could be done to strengthen the analysis:
1.     Increase the Sample Size: Larger sample sizes reduce the standard error of the mean, which would result in a larger test statistic for the same difference in means, making it more likely to reject the null hypothesis.
2.     Increase the Significance Level: If the supplier is willing to accept a higher risk of a Type I error (rejecting a true null hypothesis), they could increase the significance level to $\alpha = 0.10$ or $\alpha = 0.01$ . This would lower the critical value for the Z-test and increase the chance of rejecting the null hypothesis.
3.     Use a Two-Tailed Test: If the supplier believes there is any change in the mean hardness (not necessarily greater), a two-tailed test could be more appropriate. However, this would change the rejection criteria, and the test would need to account for deviations in both directions.
4.     Examine the Sampling Process: Ensure that the sampling process is truly random and representative of the entire population of castings, as biases in the sample could lead to misleading conclusions.
Step 8: Further Exploration: Practical Considerations
The purchase manager may also consider practical aspects beyond statistical significance when interpreting the result:
·        Economic Significance: Even if the difference in means is not statistically significant, the purchase manager might still find the difference of 0.25 units of hardness to be economically significant, depending on the industry and the impact of casting hardness on product performance.
·        Quality Control Considerations: If the hardness of castings impacts the performance or durability of the final product, then even small differences in hardness could be critical, especially in industries like aerospace, automotive, or heavy machinery manufacturing.
·        Long-Term Trends: If the purchase manager conducts periodic sampling, they could monitor whether the sample mean hardness of the supplier’s castings changes over time. This would help assess whether the supplier is gradually improving the quality of the castings, even if the difference is not statistically significant in the current sample.
Step 9: Conclusion and Final Thoughts
In conclusion, the hypothesis test indicates that, based on the sample data, the supplier's claim that his castings have a higher hardness than the historical average of 20.25 is not supported at the 5% significance level. The calculated Z-score of 1 does not exceed the critical value of 1.645, leading to a failure to reject the null hypothesis.
The purchase manager can use this information to make informed decisions about whether to continue sourcing from this supplier or to explore other suppliers who may meet the hardness specifications. However, it is important to consider that statistical results are based on sample data and inherent variability. Therefore, conducting further analysis, increasing the sample size, or considering other quality metrics could provide a more comprehensive understanding of the supplier’s capabilities.

Step 3: Determining the Critical Value

Step 4: Decision Rule

Step 5: Conclusion

Step 6: Power of the Test

Step 7: Reconsidering the Analysis

Step 8: Further Exploration: Practical Considerations

Step 9: Conclusion and Final Thoughts

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