Chapter 11: Hypothesis Tests and Estimation for Population Variance

Introduction

This chapter focuses on statistical methods for testing and estimating population variances, which are crucial for quality control and business decision-making. The main tools discussed are the chi-square test for a single population variance and the F-test for comparing two population variances.

Variance: Definition and Importance

What is Variance?

Variance is a fundamental measure in statistics that quantifies how far each number in a data set is from the mean (average), and thus from every other number in the set.

• Definition: Variance is the average of the squared differences from the mean.

• Formula:

• Application: Used to assess consistency and quality in manufacturing, service times, and other business processes.

Why Use Variance Instead of Standard Deviation?

• Statistical Tests: Most tests are formulated for variance (), not standard deviation ().

• Mathematical Properties: Variance allows for easier manipulation in equations, as addition and subtraction are not straightforward with square roots.

Hypothesis Testing for Single Population Variance

Chi-Square Test for Variance

The chi-square test is used to determine if the variance of a population differs from a specified value. It is based on the chi-square distribution, which is right-skewed and starts at zero.

• Assumptions: The population is normally distributed.

• Test Statistic:

• Degrees of Freedom:

Chi-Square Distribution Characteristics

• Shape: Right-skewed, never negative.

• Degrees of Freedom:

Hypothesis Testing Steps

1. State the Null and Alternate Hypotheses:

2. Select a Level of Significance: Common values are 0.05 or 0.10.

3. Identify the Test Statistic: Use the chi-square formula above.

4. Formulate the Decision Rule: Compare the test statistic to the critical value from the chi-square table.

5. Take a Sample and Arrive at a Decision: If the test statistic falls in the rejection region, reject .

6. Interpret the Results: State whether there is evidence to support a change in variance.

Example: Valley Appliance Repair

• Population standard deviation: 30 minutes ()

• Sample standard deviation: 35 minutes ()

• Sample size: 20 ()

• Null hypothesis:

• Test statistic:

• Critical value (right tail, , ): 27.20

• Decision: Test statistic is not in the rejection region. Fail to reject the null hypothesis.

Decision Rules for Hypothesis Tests

Types of Tests

• Left-Tail: Reject if

• Right-Tail: Reject if

• Two-Tail: Reject if or

Confidence Intervals for Population Variance

Confidence Interval Formula

Confidence intervals estimate the range in which the true population variance is likely to fall, based on sample data.

• Formula:

• Interpretation: With a specified confidence level (e.g., 95%), the true variance lies within the calculated interval.

Example: Small Theater Snack Bar Sales

• Sample size: 14

• Sample standard deviation: $38.60

• Sample variance:

• Degrees of freedom: $13$

• Confidence interval for variance:

• Confidence interval for standard deviation:

Hypothesis Tests for Two Population Variances

F-Test for Comparing Two Variances

The F-test is used to compare the variances of two independent samples to determine if they are significantly different.

• Assumptions: Both populations are normally distributed; samples are independent.

• Test Statistic: (where is the larger sample variance)

• Degrees of Freedom: ,

Example: Fast Food Drive-Thru Wait Times

• System 1: , seconds,

• System 2: , seconds,

• Test statistic:

• Critical value (two-tailed, , , ): 2.465

• Decision: Test statistic is not in the rejection region. Fail to reject the null hypothesis.

Summary Table: Hypothesis Test for Variance

Key Formulas

• Chi-Square Test Statistic:

• F-Test Statistic:

• Confidence Interval for Variance:

Conclusion

Testing and estimating population variances are essential for business statistics, especially in quality control and process improvement. The chi-square and F-tests provide rigorous methods for evaluating variability and making informed decisions based on data.