91Ó°ÊÓ

Mindy Fernandez is in charge of production at the new sport-utility vehicle (SUV) assembly plant that just opened in her town. Lately she has been concerned that the wheel lug studs do not match the chrome lug nuts close enough to keep the assembly of the wheels operating smoothly. Workers are complaining that cross-threading is happening so often that threads are being stripped by the air wrenches and that torque settings also have to be adjusted downward to prevent stripped threads even if the parts match up. In an effort to determine whether the fault lies with the lug nuts or the studs, Mindy decided to ask the quality-control department to test a random sample of 60 lug nuts and 40 studs to see if the variances in threads are the same for both parts. The report from the technician indicated that the thread variance of the sampled lug nuts was 0.00213 and that the thread variance for the sampled studs was 0.00166. What can Mindy conclude about the equality of the variances at the 0.05 level of significance? a. Solve using the \(p\) -value approach. b. Solve using the classical approach.

Step by step solution

01

Set up the null and alternate hypotheses

The null hypothesis (\(H_0\)) is that the two population variances are equal, which can be stated as \(σ^2_{nuts}=σ^2_{studs}\). The alternate hypothesis (\(H_1\)) is that the variances are not equal, which can be stated as \(σ^2_{nuts}≠ σ^2_{studs}\).

02

Calculate the F statistic

The F statistic is calculated by dividing the larger variance by the smaller variance. Our calculated variance for nuts was 0.00213 and for studs it was 0.00166. Thus, our F statistic will be \(F = \frac{0.00213}{0.00166} = 1.28\). Since the samples were random and the populations are normally distributed, F distribution applies here.

03

Calculate the critical F value

The critical F-value is obtained from the F-Distribution table, using degrees of freedom for the numerator and the denominator and level of significance which is 0.05 in this case. The degrees of freedom for the nuts (the numerator) is the sample size of nuts minus 1, or \(df_{nuts} =60-1 =59\). The degrees of freedom for the stud (the denominator) is the sample size for the stud minus 1, or \(df_{studs} = 40-1=39\). F critical value is obtained by looking up these degrees of freedom and level of significance in the F-Distribution table.

04

Make decision - p-value approach

For the p-value approach, make a decision whether to reject the null hypothesis by comparing the p-value to the level of significance. If the p-value is less than the level of significance, reject the null hypothesis. Using F-Distribution table or a statistical software, and our calculated F-statistic and degrees of freedom, we calculate the p-value of this distribution.

05

Make decision - classical approach

For the classical approach, make a decision based on whether the F statistic falls within the critical region defined by the critical F value. If the F statistic is greater than the critical F value, reject the null hypothesis. Using the F value calculated in Step 2 and the critical F value calculated in Step 3, we make this comparison.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis

The null hypothesis ( H_0 ) serves as a foundational assumption in hypothesis testing. It's the statement we aim to test, generally one of no effect or no difference. For Mindy's problem, the null hypothesis is that the variances of the lug nuts and studs are equal. This can be mathematically expressed as σ^2_{nuts}=σ^2_{studs} .
This hypothesis sets the expectation that any observed difference between the two variances is due to random fluctuation. Rejecting the null implies that the evidence suggests a real difference between variances. Otherwise, we do not find sufficient evidence against the null, indicating the variances may well be the same. The null hypothesis acts as a default or starting point for statistical testing.

F Distribution

The F distribution is crucial for analyzing variance. It's specific to tests that involve comparing two variances, commonly used in ANOVA and variance ratio tests like Mindy's. The F distribution arises from calculating the ratio of two independent chi-squared variables and is used here to determine the F statistic's probability.
For Mindy's test comparing lug nuts and studs, the F statistic is computed by dividing the larger sample variance by the smaller one. The resulting statistic falls under the F Distribution, with degrees of freedom derived from sample sizes of the two groups. Understanding the shape and characteristics of the F distribution helps in interpreting results and determining critical values for testing the null hypothesis.

Variance Analysis

Variance analysis involves comparing variability across different sets of data. This method helps identify factors contributing to discrepancies, like assembly errors. In Mindy's case, the thread variance determines the consistency of lug nuts and studs, critical for smooth wheel assembly.
Variance is calculated as the average of the squared differences from the mean. Mindy's analysis seeks to find out if variances are statistically different using the F test. By comparing computed variances, if one is significantly higher, it suggests inconsistencies in manufacturing or fit. Understanding variance analysis can provide insightful quality control measures.

Critical Value Approach

The critical value approach is a method used in hypothesis testing to make decisions about the null hypothesis. After calculating the F statistic, Mindy needs to compare it to a critical value from the F distribution table, based on her level of significance, 0.05, and degrees of freedom.
If the computed F statistic exceeds this critical F value, it indicates the result is significant at the 0.05 level, leading to a rejection of the null hypothesis. It suggests that the variances are not equal. This method highlights the statistical boundary or threshold beyond which the null is not supported by the data.

p-value Approach

The p-value approach offers a different angle in hypothesis testing, providing a probability measure. It reflects the likelihood of observing the test statistic or something more extreme by chance if the null holds true.
In Mindy's scenario, after calculating the F statistic, she can determine the p-value using statistical software or F distribution tables. This p-value is then compared against the significance level of 0.05. If the p-value is lower, it indicates strong evidence against the null hypothesis, leading to its rejection. This approach provides a direct measure of evidence, quantifying the data's support for or against the null hypothesis.