"Performing a Two-Sample F-Test In Exercises 19–26, (a) identify the claim and state H0 and Ha, (b) find the critical value and identify the rejection region, (c) find the test statistic F, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the samples are random and independent, and the populations are normally distributed.


Life of Appliances Company A claims that the variance of the lives of its appliances is less than the variance of the lives of Company B’s appliances. A sample of the lives of 20 of Company A’s appliances has a variance of 1.8. A sample of the lives of 25 of Company B’s appliances has a variance of 3.9. At α=0.025, can you support Company A’s claim?"

Verified step by step guidance

1. Step 1: Identify the claim and state the null and alternative hypotheses. The claim is that the variance of the lives of Company A's appliances is less than the variance of Company B's appliances. This translates to the alternative hypothesis (Ha): σ₁² < σ₂², where σ₁² is the variance of Company A's appliances and σ₂² is the variance of Company B's appliances. The null hypothesis (H₀) is the opposite: σ₁² ≥ σ₂².
2. Step 2: Determine the critical value and rejection region. Since this is a two-sample F-test, the test statistic follows an F-distribution. The degrees of freedom for the numerator (df₁) are n₁ - 1, where n₁ is the sample size for Company A. The degrees of freedom for the denominator (df₂) are n₂ - 1, where n₂ is the sample size for Company B. Using the significance level α = 0.025 and the F-distribution table, find the critical value for a one-tailed test. The rejection region is F < critical value.
3. Step 3: Calculate the test statistic F. The formula for the F-test statistic is F = (s₁² / s₂²), where s₁² is the sample variance of Company A's appliances and s₂² is the sample variance of Company B's appliances. Plug in the given variances (s₁² = 1.8 and s₂² = 3.9) to compute the test statistic.
4. Step 4: Compare the test statistic to the critical value. If the test statistic falls in the rejection region (F < critical value), reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
5. Step 5: Interpret the decision in the context of the original claim. If the null hypothesis is rejected, it supports the claim that the variance of the lives of Company A's appliances is less than the variance of Company B's appliances. If the null hypothesis is not rejected, there is insufficient evidence to support the claim.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which represents a statement of no effect or no difference, and the alternative hypothesis (Ha), which represents the claim being tested. In this case, H0 states that the variance of Company A's appliances is greater than or equal to that of Company B's, while Ha claims it is less.

F-Test for Variances

The F-test is a statistical test used to compare the variances of two populations to determine if they are significantly different. It calculates the F-statistic by taking the ratio of the two sample variances. In this scenario, the F-statistic will help assess whether the variance of Company A's appliances is indeed less than that of Company B's, as claimed.

Critical Value and Rejection Region

The critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. It is derived from the chosen significance level (α) and the distribution of the test statistic. The rejection region is the range of values for the test statistic that leads to rejecting H0. For this problem, with α=0.025, the critical value will help identify whether the calculated F-statistic falls within the rejection region, supporting or refuting Company A's claim.