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.
- 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=nσxˉ−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=1002.520.50−20.25
First,
calculate the denominator:
2.5100=2.510=0.25\frac{2.5}{\sqrt{100}}
= \frac{2.5}{10} = 0.251002.5=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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