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 test whether the claim of the supplier that their castings have heavier hardness is tenable, we can perform a hypothesis test using the data provided. The process of hypothesis testing is a statistical procedure used to make inferences about population parameters based on sample data. In this case, the population parameter of interest is the mean hardness of the castings, and the sample data comes from a sample of 100 castings taken from a supplier.

Step 1: Understand the Problem and Formulate Hypotheses

We are given the following information:

  • The population mean hardness (μ\muμ) of castings from any supplier is 20.25.
  • The population standard deviation (σ\sigmaσ) is 2.5.
  • The sample size (nnn) is 100.
  • The sample mean hardness (xˉ\bar{x}xˉ) is 20.50.

The supplier claims that their castings have a heavier hardness, which suggests that the mean hardness of the castings they provide is greater than 20.25. We need to test whether this claim is statistically supported by the sample data.

The hypotheses for this test are:

  • Null Hypothesis (H0H_0H0​): The population mean hardness is equal to 20.25, i.e., μ=20.25\mu = 20.25μ=20.25.
  • Alternative Hypothesis (HaH_aHa​): The population mean hardness is greater than 20.25, i.e., μ>20.25\mu > 20.25μ>20.25.

This is a one-tailed (right-tailed) test because we are testing whether the supplier's castings are significantly harder, i.e., whether the mean hardness is greater than the claimed value of 20.25.

Step 2: Set the Significance Level

In hypothesis testing, we need to set a significance level (α\alphaα) to determine the threshold for rejecting the null hypothesis. The significance level represents the probability of rejecting the null hypothesis when it is actually true (Type I error). A commonly used value for α\alphaα is 0.05, which corresponds to a 5% chance of incorrectly rejecting the null hypothesis.

Therefore, we will use α=0.05\alpha = 0.05α=0.05 for this test, meaning that if the probability of observing a sample mean as extreme or more extreme than the one observed (given that the null hypothesis is true) is less than 5%, we will reject the null hypothesis.

Step 3: Calculate the Test Statistic

To perform the hypothesis test, we need to calculate the test statistic, which in this case is the z-statistic. The formula for the z-statistic when testing a sample mean is:

z=xˉ−μσnz = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}z=n​σ​xˉ−μ​

Where:

  • xˉ\bar{x}xˉ is the sample mean (20.50),
  • μ\muμ is the population mean under the null hypothesis (20.25),
  • σ\sigmaσ is the population standard deviation (2.5),
  • nnn is the sample size (100).

We can now substitute the given values into the formula:

z=20.50−20.252.5100z = \frac{20.50 - 20.25}{\frac{2.5}{\sqrt{100}}}z=100​2.5​20.50−20.25​

First, calculate the denominator:

2.5100=2.510=0.25\frac{2.5}{\sqrt{100}} = \frac{2.5}{10} = 0.25100​2.5​=102.5​=0.25

Now, calculate the numerator:

20.50−20.25=0.2520.50 - 20.25 = 0.2520.50−20.25=0.25

Thus, the z-statistic is:

z=0.250.25=1z = \frac{0.25}{0.25} = 1z=0.250.25​=1

Step 4: Find the Critical Value

Since we are performing a one-tailed test with a significance level of α=0.05\alpha = 0.05α=0.05, we need to find the critical value of the z-statistic that corresponds to a cumulative probability of 0.95 (since we are testing for values greater than 20.25). This means we are looking for the z-score such that 95% of the distribution lies to the left of it.

Using standard z-tables or statistical software, the critical value of z for α=0.05\alpha = 0.05α=0.05 in a one-tailed test is z = 1.645. This is the cutoff value beyond which we would reject the null hypothesis.

Step 5: Make the Decision

Now, we compare the calculated z-statistic with the critical value:

  • Calculated z-statistic = 1
  • Critical z-value = 1.645

Since the calculated z-statistic (1) is less than the critical z-value (1.645), we fail to reject the null hypothesis. In other words, the sample data does not provide enough evidence to support the supplier's claim that their castings have a significantly higher hardness than the mean of 20.25.

Step 6: Conclusion

Based on the hypothesis test, we fail to reject the null hypothesis. This means that the sample data does not provide strong enough evidence to support the supplier's claim that their castings have a higher hardness. The observed sample mean of 20.50 is not significantly greater than the population mean of 20.25, considering the given sample size and standard deviation.

Therefore, we conclude that the claim of the supplier that their castings have heavier hardness is not tenable at the 5% significance level.

Step 7: Type I and Type II Errors

It is important to understand the potential errors that could occur in hypothesis testing. There are two types of errors:

1.    Type I Error: This occurs when we reject the null hypothesis when it is actually true. In this case, a Type I error would mean concluding that the supplier's castings are significantly harder when, in reality, they are not.

2.    Type II Error: This occurs when we fail to reject the null hypothesis when it is actually false. In this case, a Type II error would mean concluding that the supplier's castings are not significantly harder when, in fact, they are.

The probability of a Type I error is equal to the significance level (α\alphaα), which in this case is 0.05. The probability of a Type II error depends on factors such as the true mean hardness of the castings and the sample size.

Step 8: Additional Considerations

While the hypothesis test shows that the supplier's claim is not supported by the sample data, it is important to consider the practical significance of the result. Even though the statistical test failed to reject the null hypothesis, the difference between the population mean (20.25) and the sample mean (20.50) is 0.25. Depending on the context, this difference might or might not be practically significant. For example, if the castings are being used in a high-precision engineering application where even small differences in hardness are important, a difference of 0.25 could be meaningful. However, if the application tolerates some variation in hardness, this difference might not be significant in practical terms.

Step 9: Recommendations

Given the results of the hypothesis test, the purchase manager might want to reconsider the supplier’s claim and seek additional data or conduct further testing to verify the quality and consistency of the castings. It may also be helpful to explore other suppliers to compare the hardness of their castings.

The purchase manager could consider requesting more samples from the supplier and perform a larger-scale study to increase the power of the test. A larger sample size would reduce the standard error and could lead to a more conclusive result.

If the purchase manager is still concerned about the supplier's casting hardness, they may consider other quality control measures, such as on-site inspections, third-party testing, or supplier certification, to ensure that the castings meet the required specifications.

Final Thoughts

In conclusion, hypothesis testing provides a structured way to evaluate claims made by suppliers and other stakeholders. In this case, the test did not find sufficient evidence to support the supplier’s claim that their castings have a higher hardness than the average hardness of 20.25. The process of hypothesis testing not only helps in decision-making but also provides a framework for assessing the reliability of claims and making data-driven choices.

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