Table of contents

1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 22m
11. Correlation1h 6m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

14. ANOVA

Introduction to ANOVA

Statistics

14. ANOVA

Introduction to ANOVA

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1:57 minutes

Problem 10.RE.16

Textbook Question

In Exercises 13–16, find the critical F-value for a two-tailed test using the level of significance α and degrees of freedom d.f.N and d.f.D.

α=0.01,d.f.N=11,d.f.D=13

Verified step by step guidance

Step 1: Understand the problem. You are tasked with finding the critical F-value for a two-tailed test. The level of significance (α) is 0.01, and the degrees of freedom for the numerator (d.f.N) is 11, while the degrees of freedom for the denominator (d.f.D) is 13.

Step 2: Recall that the F-distribution is used in hypothesis testing, and the critical F-value is determined based on the level of significance (α), the degrees of freedom for the numerator (d.f.N), and the degrees of freedom for the denominator (d.f.D). For a two-tailed test, the significance level is split equally between the two tails (α/2 for each tail).

Step 3: Use an F-distribution table or statistical software to find the critical F-value. Locate the row corresponding to d.f.N = 11 and the column corresponding to d.f.D = 13. Since this is a two-tailed test, you will need to find the critical F-value for α/2 = 0.005 in each tail.

Step 4: If using an F-distribution table, ensure you are looking at the correct table for the desired significance level (α/2 = 0.005). If the exact degrees of freedom are not listed, interpolation may be required to estimate the critical F-value.

Step 5: Once the critical F-value is identified, interpret it as the threshold beyond which the test statistic would fall in the rejection region for the null hypothesis. This value will be used to determine whether to reject or fail to reject the null hypothesis in the context of the hypothesis test.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Critical F-value

The critical F-value is a threshold used in hypothesis testing to determine whether to reject the null hypothesis. It is derived from the F-distribution, which is used when comparing variances between two groups. The critical value is based on the chosen significance level (α) and the degrees of freedom for the numerator (d.f.N) and denominator (d.f.D). If the calculated F-statistic exceeds this critical value, the null hypothesis is rejected.

Degrees of Freedom

Degrees of freedom (d.f.) refer to the number of independent values or quantities that can vary in an analysis without violating any constraints. In the context of an F-test, d.f.N represents the degrees of freedom associated with the numerator (the group with more variability), while d.f.D represents the degrees of freedom for the denominator (the group with less variability). These values are crucial for determining the shape of the F-distribution and finding the critical F-value.

Level of Significance (α)

The level of significance (α) is the probability of rejecting the null hypothesis when it is actually true, also known as a Type I error. It is a threshold set by the researcher, commonly at values like 0.01 or 0.05, which indicates the likelihood of observing a test statistic as extreme as the one calculated, under the null hypothesis. In this case, α=0.01 means there is a 1% risk of concluding that a difference exists when there is none.

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Related Practice

Multiple Choice

A school administrator wants to examine whether students' academic performance differs based on the type of instructional method used in their classes. A random sample of $18$ students is selected and divided evenly among the three teaching methods. After a semester, all students take the same standardized final exam. An ANOVA test is performed and results in a P-value of $1.403 ∙ 10^{- 7}$ . Interpret these results.

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Multiple Choice

A regional sales director wants to determine whether different customer service training programs lead to different levels of employee performance across three branches. Each branch uses one of the following training programs: Program A. Program B, or Program C. After one month, the director measures the performance score (out of 100) for 5 randomly selected employees from each branch. Using $alpha equals 0.05$ , perform a one-way ANOVA to determine whether there is a statistically significant difference in mean performance among the three training programs.

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Textbook Question

In Exercises 9–12, find the critical F-value for a right-tailed test using the level of significance α and degrees of freedom d.f.N and d.f.D.

α=0.05,d.f.N=20,d.f.D=25

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Textbook Question

In Exercises 13–16, find the critical F-value for a two-tailed test using the level of significance α and degrees of freedom d.f.N and d.f.D.

α=0.01,d.f.N=40,d.f.D=60

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Textbook Question

State the null and alternative hypotheses for a one-way ANOVA test.

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