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Ch. 9 - Inferences from Two Samples
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoTriola 14th EditionCh. 9 - Inferences from Two SamplesProblema 9.4.1c
Capitolo 9, Problema 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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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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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
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c. Use a 0.05 significance level to test the claim that the two population proportions are equal. What do you conclude?


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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.


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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.


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