/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 80 Under what assumptions may the \... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 80

Under what assumptions may the \(F\) distribution be used in making inferences about the ratio of population variances?

Short Answer

Expert verified
Use the F distribution if populations are normally distributed, samples are independent and random.

Step by step solution

01

Understanding the F Distribution

The F-distribution is a ratio of two chi-squared variables divided by their respective degrees of freedom. It is used to compare variances between two populations. Understanding the basis of the F-distribution helps in knowing the conditions under which it can be applied.
02

Assumption of Normality

One of the primary assumptions for using the F-distribution to make inferences about the ratio of population variances is that the populations from which the samples are drawn must be normally distributed. This means that each population must follow a normal distribution.
03

Assumption of Independence

The samples taken from the two populations must be independent of each other. This ensures that the variation in one sample does not affect or correlate with the variation in the other sample.
04

Assumption of Random Sampling

The samples must be randomly selected from their respective populations. Random sampling ensures that the sample is a true representation of the population, giving validity to the results of the F-test.
05

Equal Variances within Samples

Each set of data (for the two populations being compared) should individually have variances assumed to be equal, which refers to the assumption of homogeneity of variance within each sample, not between the samples.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Normality Assumption
The normality assumption is crucial when using the F-distribution to make inferences about the ratio of population variances. This assumption means that the data sampled from each population must follow a normal distribution.
This is important because the F-test is based on the properties of the normal distribution, which allows it to accurately determine the ratio of variances. If the populations do not meet this requirement, the F-test results may not be valid.
  • Normality provides a predictable pattern.
    It ensures the robustness of the test.
  • The test's conclusions are more reliable when normality is met.
Tools such as QQ plots or the Shapiro-Wilk test can help in testing normality before applying the F-test.
Independent Samples
When applying the F-distribution, it's vital to ensure that the samples under study are independent. This means the data from one sample should not influence or have any correlation with the data from the other sample.
Why is this so important? Because independence ensures that the comparisons made between variances are valid.
  • An independent sample maintains the purity of results.
    Interference between samples can lead to biased results.
  • Ensuring independence contributes to the methodical strength of statistical analysis.
In practice, independence is often assured by clear separate experimental processes or well-designed randomization.
Random Sampling
Random sampling is a cornerstone of statistical analysis. When using the F-distribution, ensuring that samples are randomly chosen from the populations they represent is essential. This sampling method helps to ensure that the sample reflects the larger population.
With random sampling, the chance of bias is significantly decreased, making the inference more authentic.
  • It minimizes sampling bias, giving every member of the population an equal chance of selection.
  • Random sampling strengthens the argument that sample findings are applicable to the entire population.
Before conducting an F-test, check if random sampling was employed to maintain the integrity of the results.
Homogeneity of Variance
Homogeneity of variance implies that within each group being compared, the variances should be equal. While the F-test compares variances between groups, it assumes each group's internal variance is consistent.
Why is this assumption important? Because variability within individual samples should be stable for the comparison to focus on variance differences between groups.
  • Equal variance within samples ensures the accuracy of the F-test.
  • If the assumption is violated, the test may not accurately reflect the population differences.
Consider conducting tests like Levene's test to verify the homogeneity assumption before relying on the F-test results.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Let \(Y_{1}, Y_{2}, \ldots, Y_{n}\) denote a random sample from a population having a Poisson distribution with mean \(\lambda\) a. Find the form of the rejection region for a most powerful test of \(H_{0}: \lambda=\lambda_{0}\) against \(H_{a}: \lambda=\lambda_{a}\) where \(\lambda_{a}>\lambda_{0}\) b. Recall that \(\sum_{i=1}^{n} Y_{i}\) has a Poisson distribution with mean \(n \lambda\). Indicate how this information can be used to find any constants associated with the rejection region derived in part (a). c. Is the test derived in part (a) uniformly most powerful for testing \(H_{0}: \lambda=\lambda_{0}\) against \(H_{a}:\) \(\lambda>\lambda_{0} ?\) Why? d. Find the form of the rejection region for a most powerful test of \(H_{0}: \lambda=\lambda_{0}\) against \(H_{a}: \lambda=\lambda_{a}\) where \(\lambda_{a}<\lambda_{0}\)

Suppose that \(Y_{1}, Y_{2}, \ldots, Y_{n}\) constitute a random sample from a normal distribution with known mean \(\mu\) and unknown variance \(\sigma^{2}\). Find the most powerful \(\alpha\) -level test of \(H_{0}: \sigma^{2}=\sigma_{0}^{2}\) versus \(H_{a}:\) \(\sigma^{2}=\sigma_{1}^{2},\) where \(\sigma_{1}^{2}>\sigma_{0}^{2} .\) Show that this test is equivalent to a \(\chi^{2}\) test. Is the test uniformly most powerful for \(H_{a}: \sigma^{2}>\sigma_{0}^{2} ?\)

What assumptions are made when a Student's \(t\) test is employed to test a hypothesis involving a population mean?

An article in American Demographics reports that \(67 \%\) of American adults always vote in presidential elections. \(^{\star}\) To test this claim, a random sample of 300 adults was taken, and 192 stated that they always voted in presidential elections. Do the results of this sample provide sufficient evidence to indicate that the percentage of adults who say that they always vote in presidential elections is different than the percentage reported in American Demographics? Test using \(\alpha=.01\)

Currently, 20\% of potential customers buy soap of brand A. To increase sales, the company will conduct an extensive advertising campaign. At the end of the campaign, a sample of 400 potential customers will be interviewed to determine whether the campaign was successful. a. State \(H_{0}\) and \(H_{a}\) in terms of \(p\), the probability that a customer prefers soap brand A. b. The company decides to conclude that the advertising campaign was a success if at least 92 of the 400 customers interviewed prefer brand \(A\). Find \(\alpha\). (Use the normal approximation to the binomial distribution to evaluate the desired probability.)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.