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.

To assess whether the claim made by the supplier regarding the hardness of their castings is tenable, we can conduct a hypothesis test. This is a common method used in statistics to determine whether there is enough evidence to support or reject a particular claim based on sample data. In this case, the supplier claims that their castings have a higher hardness than the known mean hardness of castings from any supplier, which is 20.25. The purchase manager collects 100 samples and finds that the mean hardness of the castings from this supplier is 20.50. Given the population standard deviation of 2.5, the manager can conduct a hypothesis test to determine if the sample mean hardness of 20.50 is statistically significantly higher than the population mean of 20.25.

Step 1: Formulate the Hypotheses

The first step in hypothesis testing is to clearly define the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis typically represents the status quo or the claim that we are trying to test against, while the alternative hypothesis represents the claim or situation that we are trying to prove.

· Null Hypothesis (H₀): The mean hardness of the supplier’s castings is equal to the population mean hardness, i.e., 20.25. Mathematically, H₀: μ = 20.25.

· Alternative Hypothesis (H₁): The mean hardness of the supplier’s castings is greater than the population mean hardness, i.e., 20.25. This is a one-tailed test because the supplier is claiming that the castings are "heavier" in hardness, which implies a direction (greater than). Mathematically, H₁: μ > 20.25.

Step 2: Set the Significance Level (α)

The significance level (α) represents the probability of rejecting the null hypothesis when it is true (Type I error). Commonly, a significance level of 0.05 (5%) is used in hypothesis testing. This means that there is a 5% chance of incorrectly rejecting the null hypothesis when it is actually true. For this test, we will use α = 0.05, which means we are willing to accept a 5% probability of making a Type I error.

Step 3: Calculate the Test Statistic

Since the population standard deviation (σ) is known and the sample size (n) is large (n = 100), we can use a Z-test to test the hypothesis. The formula for the Z-test statistic in this case is:

$Z=Xˉ−μ0σnZ = \frac{\bar{X} - \mu_0}{\frac{\sigma}{\sqrt{n}}}$ Z=nσXˉ−μ0

Where:

$Xˉ\bar{X}$ Xˉ is the sample mean hardness (20.50),
$μ0\mu_0$ μ0 is the population mean hardness (20.25),
$σ\sigma$ σ is the population standard deviation (2.5),
n is the sample size (100).

Plugging in the values:

$Z=20.50−20.252.5100Z = \frac{20.50 - 20.25}{\frac{2.5}{\sqrt{100}}}$ Z=1002.520.50−20.25 $Z=0.252.510Z = \frac{0.25}{\frac{2.5}{10}}$ Z=102.50.25 $Z=0.250.25=1.0Z = \frac{0.25}{0.25} = 1.0$ Z=0.250.25=1.0

The test statistic (Z) is 1.0.

Step 4: Determine the Critical Value

For a one-tailed test with a significance level of α = 0.05, we need to determine the critical value of Z. This can be done using a Z-table or statistical software. The critical value corresponds to the Z-score that marks the cutoff beyond which we will reject the null hypothesis. For a significance level of 0.05 in a one-tailed test, the critical Z-value is approximately 1.645.

Step 5: Make the Decision

Now that we have the test statistic (Z = 1.0) and the critical value (Z₁₋₀.₀₅ = 1.645), we can make our decision:

If the test statistic Z is greater than the critical value (Z > 1.645), we reject the null hypothesis.
If the test statistic Z is less than or equal to the critical value (Z ≤ 1.645), we fail to reject the null hypothesis.

In this case, the test statistic (Z = 1.0) is less than the critical value (Z = 1.645). Therefore, we fail to reject the null hypothesis.

Step 6: Conclusion

Based on the hypothesis test, we do not have sufficient evidence to reject the null hypothesis at the 0.05 significance level. The sample mean hardness of 20.50 is not statistically significantly greater than the population mean hardness of 20.25. Thus, the claim of the supplier that their castings have a higher hardness is not tenable based on this sample data.

This conclusion suggests that, from a statistical standpoint, the supplier’s claim about having castings with greater hardness than the known population mean is not supported by the sample data. However, it is important to note that this does not prove that the supplier's claim is false—only that there is insufficient statistical evidence to support it given the sample provided.

Step 7: Additional Considerations

It is important to consider a few factors that might affect the robustness of this hypothesis test:

1. Sample Size: The sample size of 100 is relatively large, which generally increases the reliability of the results. Larger samples tend to provide more accurate estimates of population parameters and reduce the variability of the sample mean. This increases the power of the hypothesis test, allowing us to detect small differences if they exist.

2. Population Standard Deviation: The test assumes that the population standard deviation is known and constant. If the population standard deviation is uncertain or if it varies, alternative methods (such as a t-test) might be more appropriate.

3. Effect Size: Even though the test did not result in a rejection of the null hypothesis, it is worth noting the size of the difference between the sample mean (20.50) and the population mean (20.25). The difference of 0.25 may seem small, but it is statistically insignificant in this case, given the sample size and the standard deviation. If the sample size were larger or if the population standard deviation were smaller, the difference might become significant.

4. Practical Significance: Statistical significance does not necessarily imply practical significance. Even if a difference is not statistically significant, it might still be practically or economically significant, depending on the context. In this case, the difference in hardness between 20.50 and 20.25 may not be large enough to have any meaningful impact on the performance or quality of the castings.

5. Type I and Type II Errors: A Type I error occurs when the null hypothesis is incorrectly rejected, while a Type II error occurs when the null hypothesis is incorrectly not rejected. In this case, failing to reject the null hypothesis (Type II error) means that we might be incorrectly concluding that the supplier’s claim is unsupported, when in fact, the castings might be harder. A larger sample size or more precise data could help reduce the risk of Type II errors.

6. Further Testing: If the supplier insists on their claim or if further investigation is warranted, additional testing with more samples could provide more robust evidence. A higher level of confidence or further analysis using different statistical methods might yield more conclusive results.

Step 8: Reflection on Hypothesis Testing

Hypothesis testing is a powerful tool for making decisions based on sample data, but it is important to understand its limitations. A failure to reject the null hypothesis does not prove that the null hypothesis is true, only that there is insufficient evidence to support the alternative hypothesis. Similarly, rejecting the null hypothesis does not prove that the alternative hypothesis is true—it only suggests that there is strong evidence for it.

In this case, the purchase manager’s test of the supplier’s claim about the hardness of their castings has not provided sufficient evidence to conclude that the castings are indeed harder than the known mean of 20.25. The manager should consider these statistical results, but also weigh other factors such as the cost of additional testing, the practical importance of the hardness difference, and the supplier’s reputation.

Conclusion

In conclusion, the statistical test based on the Z-test did not provide sufficient evidence to support the supplier's claim of higher hardness for their castings. The null hypothesis that the mean hardness is 20.25 is not rejected at the 0.05 significance level. While the sample mean of 20.50 is slightly greater than the population mean of 20.25, this difference is not statistically significant given the sample size and variability of the data. Therefore, the claim made by the supplier is not tenable based on the current evidence. However, the purchase manager may choose to conduct further investigations or gather more data to ensure that a comprehensive decision is made.