91Ó°ÊÓ

The Chi-Square Test is a statistical method used to compare and evaluate whether an observed variance significantly differs from a specified value in the population.
In our problem, this test helps to determine whether the variability in the fan blades exceeds the safe limit set by regulations.
To perform this test, you formulate two hypotheses:

• The null hypothesis (\( H_0 \)): assumes there is no effect or difference, in other words, that the variance is equal to 0.18 mm².
• The alternative hypothesis (\( H_a \)): which states that the variance is greater than 0.18 mm².

To calculate the chi-square statistic, you use the formula: \[ \chi^2 = \frac{(n-1)s^2}{\sigma^2} \] where \( n \) is the sample size, \( s^2 \) is the sample variance, and \( \sigma^2 \) is the hypothesized population variance.
This calculated value is then compared against a critical value from the chi-square distribution table to decide whether to reject the null hypothesis.
In this case, since the computed chi-square statistic exceeded the critical value (90 > 82.53), we reject the null hypothesis, supporting the inspector's claim that the variance is indeed beyond the allowable limit.

Population variance is a measure of the variability within a whole group or population. It indicates how much the individual data points differ from the mean of the data set.
A low variance suggests that the data points tend to be very close to the mean, whereas a high variance implies that the data points are spread out over a wider range of values.
In the context of this exercise, the population variance is critical for ensuring safety standards for jet engine fan blades.
The regular variability or variance allowed is \( \sigma^2 = 0.18 \, mm^2 \), which represents a safe threshold for operation.
In our case, a sample was taken to make inferences about the entire engine. The sample revealed a variance of \( 0.27 \, mm^2 \), which prompted the use of hypothesis testing to determine if this level of variability was truly different from the targeted population variance of 0.18 mm².
Using the sample variance and size, the test helps infer whether the variability meets safety criteria and whether actions, like replacement, should be taken.

The level of significance (\( \alpha \)) in hypothesis testing is a threshold set by the researcher to determine when to reject the null hypothesis.
It represents the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true.
Using a 0.01 level of significance means that there is only a 1% chance of incorrectly concluding that the population variance exceeds the safety limit when it does not.
This stricter level of significance is vital in safety-critical applications like jet engine inspections, as making incorrect conclusions could have serious consequences.
Therefore, by setting a low \( \alpha \), the inspector ensures greater confidence in the decision to replace the fan blades if the null hypothesis is rejected.
In simpler terms: at a 1% significance level, the evidence must be very strong before we decide that the population variance exceeds the safe threshold of 0.18 mm².

A confidence interval provides a range of values within which we can say, with a certain level of confidence, that the true population parameter lies.
In this context, we are interested in the confidence interval for the population standard deviation, which is more informative than just knowing if the variance exceeds a certain threshold.
Calculating a 90% confidence interval for the population standard deviation involves using the chi-square distribution: \[\left( \sqrt{\frac{(n-1)s^2}{\chi^2_{\text{upper}}}}, \sqrt{\frac{(n-1)s^2}{\chi^2_{\text{lower}}}} \right)\] Here, \(\chi^2_{\text{upper}}\) and \(\chi^2_{\text{lower}}\) are the critical values that mark the upper and lower bounds of the interval, respectively.
The computed confidence interval in our problem was approximately (0.20, 0.25) mm, indicating that we are 90% confident the true standard deviation lies within this range.
This range being higher than \( \sqrt{0.18} \approx 0.424 \text{ mm} \), further supports the decision to replace all fan blades to ensure safety.