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Chapter 10: Problem 118

Determine the \(p\) -value that would be used to test the following hypotheses when \(F\) is used as the test statistic: a. \( H_{o}: \sigma_{1}=\sigma_{2}\) versus \(H_{a}: \sigma_{1}>\sigma_{2},\) with \(n_{1}=10, n_{2}=16\) and \(F_{*}=2.47\) b. \(H_{o}: \sigma_{1}^{2}=\sigma_{2}^{2}\) versus \(H_{a}: \sigma_{1}^{2}>\sigma_{2}^{2},\) with \(n_{1}=25, n_{2}=21\) and \(F *=2.31\) c. \(H_{o}: \frac{\sigma_{1}^{2}}{\sigma_{2}^{2}}=1\) versus \(H_{a}: \frac{\sigma_{1}^{2}}{\sigma_{2}^{2}} \neq 1,\) with \(n_{1}=41, n_{2}=61\) and \(F \star=4.78\) d. \( H_{o}: \sigma_{1}=\sigma_{2}\) versus \(H_{a}: \sigma_{1}<\sigma_{2},\) with \(n_{1}=10, n_{2}=16\) and \(F_{*}=2.47\)

Short Answer

Expert verified
This task requires using an F-distribution table or software to find the p-value given the observed F-score and degrees of freedom, taking into account that it is a one-tailed test. As the answer depends on an external resource (the table or software), a specific numerical answer is not provided here.

Step by step solution

01

Study the Hypotheses

Understand the different hypotheses and what they mean.\na. Null hypothesis (\(H_0\)): There is no difference in variance between the two groups.\nb. Alternate hypothesis (\(H_a\)): The variance of the first group is more than the second group.
02

Compute Degrees of Freedom

The degrees of freedom for the numerator and denominator in the F-score computation are computed as \(n_1 - 1\) and \(n_2 - 1\) respectively.\nFor instance, in case a, degrees of freedom are \(10-1 = 9\) (numerator) and \(16-1 = 15\) (denominator). Similarly calculate for other cases.
03

Use an F-distribution Table or Software to Find the P-value

Given the computed F-score and degrees of freedom, use an F-distribution table or software to find the corresponding p-value. The p-value is the probability that you would observe an F-score as extreme as, or more extreme than, the observed value under the null hypothesis. Remember that since the hypotheses are one-tailed (greater than or less than, not equal/unequal), this is a one-tailed test.

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Key Concepts

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

F-test
The F-test is a statistical test that helps determine if there are significant differences between the variances or means of two populations.
This test is typically used when you want to compare variances to check if one is significantly larger than the other, or when comparing more than two groups simultaneously.
It's particularly useful for:
  • Comparing two sample variances to conclude if they come from populations having equal variances.
  • Assessing hypotheses about the ratios of variances between two populations.
The test involves calculating a test statistic, known as the F-statistic, which follows the F-distribution under the null hypothesis.
This F-statistic is a ratio of the two sample variances, which provides a basis for comparison.
It is important to ensure that the data meets the assumption of normality and that the samples are independent.
Null Hypothesis
The null hypothesis, denoted as \(H_0\), is a statement positing that there is no effect or no difference in the context of the experiment or comparison.
In variance testing using F-test, the null hypothesis often states that the variances of two populations are equal (e.g., \( \sigma_1^2 = \sigma_2^2 \)).
Crucial points about the null hypothesis:
  • It serves as a default or neutral statement to test against.
  • It does not reflect the existence of a particular effect or relationship.
  • The objective of testing is to find evidence to reject this hypothesis.
It acts as the starting point for statistical testing. If the test yields a significant p-value, we may have enough evidence to reject the null hypothesis in favor of the alternative.
Alternative Hypothesis
The alternative hypothesis, symbolically represented as \(H_a\), is an assertion that counters the null hypothesis.
It proposes that there is indeed a significant effect, difference, or relationship as per the context of the analysis.
In cases where variance is being tested, it often suggests that the variance of one group is greater than, less than, or not equal to that of another.
Characteristics of the alternative hypothesis include:
  • It is what the experimenter typically believes to be true or wishes to prove.
  • Acceptance or validation of this hypothesis occurs if statistical evidence strongly supports it over the null hypothesis.
  • It can be one-sided (greater than or lesser than) or two-sided (not equal).
Rejecting the null hypothesis implies that there is sufficient evidence for the alternative hypothesis, meaning there is a significant result that needs attention.
Degrees of Freedom
Degrees of freedom, often abbreviated as DOF, refers to the number of independent values that can vary in an analysis, while estimating statistical parameters.
In the context of the F-test:
  • The degrees of freedom for the numerator are given by \(n_1 - 1\), where \(n_1\) is the sample size of the first group.
  • The degrees of freedom for the denominator are given by \(n_2 - 1\), where \(n_2\) is the sample size of the second group.
Understanding the degrees of freedom is essential because:
  • It helps ensure the correct distribution is applied when evaluating the F-statistic.
  • It affects the shape and nature of the probability distribution used in hypothesis testing.
  • More degrees of freedom generally provide more reliable statistical results.
Correctly calculating the degrees of freedom is critical in obtaining valid p-values and making appropriate inferences from statistical tests.

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Most popular questions from this chapter

A soft drink distributor is considering two new models of dispensing machines. Both the Harvard Company machine and the Fizzit machine can be adjusted to fill the cups to a certain mean amount. However, the variation in the amount dispensed from cup to cup is a primary concern. Ten cups dispensed from the Harvard machine showed a variance of \(0.065,\) whereas 15 cups dispensed from the Fizzit machine showed a variance of \(0.033 .\) The factory representative from the Harvard Company maintains that his machine had no more variability than the Fizzit machine. Assume the amount dispensed is normally distributed. At the 0.05 level of significance, does the sample refute the representative's assertion? a. Solve using the \(p\) -value approach. b. Solve using the classical approach.

In a survey of 300 people from city \(A, 128\) prefer New Spring soap to all other brands of deodorant soap. In city \(\mathrm{B}, 149\) of 400 people prefer New Spring soap. Find the \(98 \%\) confidence interval for the difference in the proportions of people from the two cities who prefer New Spring soap.

An experiment is designed to study the effect diet has on uric acid level. The study includes 20 white rats. Ten rats are randomly selected and given a junk- food diet; the other 10 rats receive a high-fiber, low-fat diet. Uric acid levels of the two groups are determined. Do the resulting sets of data represent dependent or independent samples? Explain.

An insurance company is concerned that garage A charges more for repair work than garage B charges. It plans to send 25 cars to each garage and obtain separate estimates for the repairs needed for each car. a. How can the company do this and obtain independent samples? Explain in detail. b. How can the company do this and obtain dependent samples? Explain in detail.

Determine the test criteria that would be used with the classical approach to test the following hypotheses when \(t\) is used as the test statistic. a. \(\quad H_{o}: \mu_{d}=0\) and \(H_{a}: \mu_{d}>0,\) with \(n=15\) and \(\alpha=0.05\) b. \(\quad H_{o}: \mu_{d}=0\) and \(H_{a}: \mu_{d} \neq 0,\) with \(n=25\) and \(\alpha=0.05\) c. \(\quad H_{o}: \mu_{d}=0\) and \(H_{a}: \mu_{d}<0,\) with \(n=12\) and \(\alpha=0.10\) d. \(\quad H_{o}: \mu_{d}=0.75\) and \(H_{a}: \mu_{d}>0.75,\) with \(n=18\) and \(\alpha=0.01\)
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