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Ch. 9 - Inferences from Two Samples
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 9 - Inferences from Two SamplesProblem 9.4.1c
Chapter 9, Problem 9.4.1c

F Test Statistic


c. If testing the claim that sigma2,1 is not equals to sigma2,2 what do we know about the two samples if the test statistic F is very close to 1?

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1
Understand the context of the F-test: The F-test is used to compare the variances of two populations. The null hypothesis (H0) states that the variances are equal (σ²₁ = σ²₂), while the alternative hypothesis (H1) states that the variances are not equal (σ²₁ ≠ σ²₂).
Recall the formula for the F-test statistic: F = (s₁² / s₂²), where s₁² and s₂² are the sample variances of the two groups. The larger variance is typically placed in the numerator to ensure F ≥ 1.
Interpret the value of the F-test statistic: If the F-test statistic is very close to 1, it suggests that the two sample variances (s₁² and s₂²) are nearly equal, which supports the null hypothesis (H0).
Relate the F-test statistic to the hypothesis test: A value of F close to 1 means there is little evidence to reject the null hypothesis. This implies that the two samples likely come from populations with similar variances.
Conclude the interpretation: When F is close to 1, it indicates that the variability in the two samples is similar, and there is no strong evidence to suggest a significant difference in population variances.

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Key Concepts

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

F Test Statistic

The F test statistic is a ratio used in statistical hypothesis testing to compare variances between two populations. It is calculated by dividing the variance of one sample by the variance of another. A value close to 1 suggests that the variances of the two samples are similar, which is a key consideration when testing hypotheses about their equality.
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Step 2: Calculate Test Statistic

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. In this context, the null hypothesis typically states that the variances of the two populations are equal, while the alternative hypothesis posits that they are not. The outcome of the F test helps determine whether to reject or fail to reject the null hypothesis.
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Step 1: Write Hypotheses

Variance

Variance is a measure of the dispersion or spread of a set of data points in a sample or population. It quantifies how much the values differ from the mean of the dataset. In the context of the F test, comparing variances helps assess whether the variability in two different samples is statistically significant, which is crucial for understanding the relationship between the two populations being studied.
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Variance & Standard Deviation of Discrete Random Variables
Related Practice
Textbook Question

Overlap of Confidence Intervals In the article “On Judging the Significance of Differences by Examining the Overlap Between Confidence Intervals,” by Schenker and Gentleman (American Statistician, Vol. 55, No. 3), the authors consider sample data in this statement: “Independent simple random samples, each of size 200, have been drawn, and 112 people in the first sample have the attribute, whereas 88 people in the second sample have the attribute.”


c. Use a 0.05 significance level to test the claim that the two population proportions are equal. What do you conclude?


Textbook Question

Can Dogs Detect Malaria? A study was conducted to determine whether dogs could detect malaria from socks worn by malaria patients and socks worn by patients without malaria. Among 175 socks worn by malaria patients, the dogs made correct identifications 123 times. Among 145 socks worn by patients without malaria, the dogs made correct identifications 131 times (based on data presented at an annual meeting of the American Society of Tropical Medicine, by principal investigator Steve Lindsay). Use a 0.05 significance level to test the claim of no difference between the two rates of correct responses.


c. What do the results suggest about the use of dogs to detect malaria?

Textbook Question

In Exercises 5–16, use the listed paired sample data, and assume that the samples are simple random samples and that the differences have a distribution that is approximately normal.


Heights of Presidents A popular theory is that presidential candidates have an advantage if they are taller than their main opponents. Listed are heights (cm) of presidents along with the heights of their main opponents (from Data Set 22 “Presidents” in Appendix B).


b. Construct the confidence interval that could be used for the hypothesis test described in part (a). What feature of the confidence interval leads to the same conclusion reached in part (a)?

Textbook Question

Independent Samples Which of the following involve independent samples?


b. Data Set 6 “Births” includes birth weights of a sample of baby boys and a sample of baby girls.


Textbook Question

Confidence Interval Assume that we want to use the sample data in Exercise 1 for constructing a confidence interval to be used for testing the given claim.


c. If the resulting confidence interval is -5.8 admissions <ud < -0.9 admissions, what do you conclude?

Textbook Question

Can Dogs Detect Malaria? A study was conducted to determine whether dogs could detect malaria from socks worn by malaria patients and socks worn by patients without malaria. Among 175 socks worn by malaria patients, the dogs made correct identifications 123 times. Among 145 socks worn by patients without malaria, the dogs made correct identifications 131 times (based on data presented at an annual meeting of the American Society of Tropical Medicine, by principal investigator Steve Lindsay). Use a 0.05 significance level to test the claim of no difference between the two rates of correct responses.


b. Test the claim by constructing an appropriate confidence interval.