"Performing a Two-Sample F-Test In Exercises 19–26, (a) identi...

Question

"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?"

Accepted Answer

Hello, everyone, let's take a look at this question together. A pharmaceutical company claims that the variability in the effectiveness of its new drug is less than that of a competitor's drug. A sample of 15 patients using the new drug has a variance of 1.2, while a sample of 20 patients using the competitor's drug has a variance of 2.8. At alpha equals 0.01, is there enough evidence to support the company's claim? Is it answer choice A? Do not reject the null hypothesis, as there is evidence to support the claim. Answer choice B, reject the null hypothesis, as there is not enough evidence to support the claim. Answered Cho C, do not reject the null hypothesis, as there is not enough evidence to support the claim or answer choice D, reject the null hypothesis, as there is evidence to support the claim. So in order to solve this question, we have to recall how to test a claim at the alpha equals 0.01 significance level, so we can determine whether there is enough evidence to support the company's claim that the variability in the effectiveness of its new drug is less than that of a competitor's drug, given a sample of 5. Patients using the new drug with a variance of 1.2, and a sample of 20 patients using the competitor's drug with a variance of 2.8. And so the first step in determining whether there is enough evidence to support the company's claim is to identify the claim and state the null and alternative hypotheses. Which we know that the claim is that the variance in the effectiveness of the new drug is less than the variance in the effectiveness of the competitor's drug. So our null hypothesis is the variance of the new drug is greater than or equal to the variance of the competitor's drug, and our The alternative hypothesis is that the variance of the new drug is less than the variance of the competitor's drug, and then the next step is to find the critical value and the rejection region, which we know that this is a left-tailed F test with a sample size 1 of 15, a sample size 2 of 20. A degrees of freedom 1 of 14 degrees of freedom 2 of 19, and our alpha is equal to 0.01. So our critical value is given as F subalpha, degrees of freedom 1 and degrees of freedom 2, which is F 0.01, 14, and 19, which means that our left tail F critical value is equal to 1 over our F critical value. which is approximately 1 divided by 3.29, giving us our left tailed critical value of approximately 0.304. And now we can find our test statistic F which, using the variances for both samples, our test statistic F is equal to 1.2 divided by 2.8, which is equal to 0.429. And then using both the test statistic. and the critical values, we can compare our F statistic to the rejection region, which our test statistic F of 0.429 is greater than our rejection region value of 0.304. Therefore, we do not reject the null hypothesis, and we can conclude that there is not enough evidence at alpha equals 0.01 to support the claim that the new drug has less variability in effectiveness than the competitor's drug. And looking at our answer choices, we can identify that answer choice C is the correct answer, as we do not reject the null hypothesis, since there is not enough evidence to support the claim that the variability in the effectiveness of a company's new drug is less than that of a competitor's drug. So answer choice C is the only correct answer choice. I hope you found this video to be helpful. Thank you and goodbye.

Key Concepts

Hypothesis Testing

F-Test for Variances

Critical Value and Rejection Region

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