F-Test and Hypothesis Testing for Variance in U.S. Average Sales

Math Problem Statement

Find the test statistic.

F Subscript STATequals=3.363.36

(Round to two decimal places as needed.)

Part 3

Determine the p-value.

p-valueequals=0.0120.012

(Round to three decimal places as needed.)

Part 4

Reach a decision.

Reject

H Subscript 0.

There is

sufficient

evidence at the

0.050.05

level of significance of a difference in the mean U.S. average sales per unit among the food segments.

Part 5

b. At the

0.050.05

level of significance, is there a difference in the variation in U.S. average sales per unit ($ thousands) among the food segments? Determine the hypotheses. Choose the correct answer below.

H Subscript 0:

sigma Subscript 1 Superscript 2σ21equals=sigma Subscript 2 Superscript 2σ22equals=times•times•times•equals=sigma Subscript 48 Superscript 2σ248

H Subscript 1:

Not all

sigma Subscript j Superscript 2σ2j

are equal

(where

jequals=1,2,...,4848)

H Subscript 0:

sigma Subscript 1 Superscript 2σ21equals=sigma Subscript 2 Superscript 2σ22equals=times•times•times•equals=sigma Subscript 6 Superscript 2σ26

H Subscript 1:

Not all

sigma Subscript j Superscript 2σ2j

are equal

(where

jequals=1,2,...,66)

Your answer is correct.

H Subscript 0:

sigma Subscript 1 Superscript 2σ21equals=sigma Subscript 2 Superscript 2σ22equals=times•times•times•equals=sigma Subscript 6 Superscript 2σ26

H Subscript 1:

sigma Subscript 1 Superscript 2σ21not equals≠sigma Subscript 2 Superscript 2σ22not equals≠times•times•times•not equals≠sigma Subscript 6 Superscript 2σ26

H Subscript 0:

sigma Subscript 1 Superscript 2σ21equals=sigma Subscript 2 Superscript 2σ22equals=times•times•times•equals=sigma Subscript 48 Superscript 2σ248

H Subscript 1:

sigma Subscript 1 Superscript 2σ21not equals≠sigma Subscript 2 Superscript 2σ22not equals≠times•times•times•not equals≠sigma Subscript 48 Superscript 2σ248

Part 6

Find the test statistic.

F Subscript STATequals=enter your response here

(Round to two decimal places as needed.)

Solution

Part 5: Hypothesis for Variance in U.S. Average Sales

In this step, you're testing whether the variances in U.S. average sales per unit differ across food segments. The correct set of hypotheses is about comparing the variances ( $\sigma_j^2$ ) among all groups. This is known as a test of equal variances, often performed using an F-test or a more general method like Levene's test.

The answer that properly matches this situation is:

Answer B:

$H_0$ : $\sigma_1^2 = \sigma_2^2 = \dots = \sigma_6^2$

(All group variances are equal.)
$H_1$ : Not all $\sigma_j^2$ are equal.

(At least one group has a variance different from others.)

Part 6: Finding the Test Statistic

To find the test statistic $F_{\text{STAT}}$ , you will need the data for each group’s variance. The F-test compares the ratio of the two sample variances (or more in the case of multiple groups), usually computed as:

$F_{\text{STAT}} = \frac{\text{Largest variance}}{\text{Smallest variance}}$

Without the specific group variances provided in the question, it is impossible to calculate the exact $F_{\text{STAT}}$ value. However, you would round the computed F-statistic to two decimal places as needed.

Would you like assistance with calculating the test statistic, or have any other details about the variances to provide?

Follow-up Questions:

What are the exact variances of each group, so we can compute the $F_{\text{STAT}}$ ?
How do you calculate the $F$ -statistic for comparing variances?
What does rejecting $H_0$ mean in the context of variance?
Can you explain the difference between $H_0$ and $H_1$ for a variance test?
What happens if the test statistic exceeds the critical value?

Tip:

When performing an F-test, always ensure the sample sizes are approximately equal, or the results might be skewed.

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Math Problem Analysis

Mathematical Concepts

Statistics
Hypothesis Testing
ANOVA
F-Test
p-value

Formulas

F_STAT = Largest variance / Smallest variance
p-value calculation based on F-distribution

Theorems

F-distribution
Hypothesis Testing in ANOVA

Suitable Grade Level

Grades 11-12, Undergraduate

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