In Exercises 15–22, test the claim about the population varian...

Question

In Exercises 15–22, test the claim about the population variance or standard deviation at the level of significance Assume the population is normally distributed.

Claim: σ^2>19, α=0.1. Sample statistics: s^2=28, n=17

Accepted Answer

Hello, everyone, let's take a look at this question together. A researcher claims that the population variance of exam scores is greater than 16. A sample of N equals 12 students yields a sample variance of 24. Test the claim at the 0.10 significance level, assuming normality. What is the correct conclusion? Is it answer choice A, there is no sufficient evidence at alpha equals 0.1 to support the claim that the population variance is greater than 16. Answer choice B, there is sufficient evidence at alpha equals 0.1 to support the claim that the population variance is greater than 16, or answer choice C, not enough information. So in order to solve this question, we have to recall how we can test a claim, so that we can test the claim that the population variance of exam scores is greater than 16 at the 0.10 significance level, given that we have a sample size N of 12 and Sample variance of 24, and we must also assume normality and we know that the first step in testing this claim is to state the hypotheses, starting with the null hypothesis, which is that the population variance is less than or equal to 16, and then we have our alternative hypothesis, which is that the population variance is greater than 16. And then our next step is to compute the test statistic. Which we know we use the chi square test statistic for standard deviation, and plugging in the values into our chi square test statistic, we have chi square is equal to 12 minus 1 in parentheses, multiplied by 24, divided by 16. Which is equal to 16.5, so our chi square test statistic value is 16.5. And then we find the critical value, which we are looking for, the critical value for an alpha equals 0.1, and a degrees of freedom equaling 11, as well as this is a right tail test. So from the chi square distribution tables, Our chi square critical value is approximately 17.275. And since we have determined our test statistic and our chi square critical value, we can compare the test statistic to the critical value. Which we know 16.5 is less than 17.275, and since the test statistic is less than the critical value, we failed to reject the null hypothesis. And so we can conclude that there is no sufficient evidence at alpha equals 0.1 to support the claim that the population variance is greater than 16. Which, looking at our answer choices, we can identify answer choice A is the correct answer. As we know, there is no sufficient evidence at alpha equals 0.1 to support the claim that the population variance of the exam scores is greater than 16, and all other answer choices are incorrect. I hope you found this video to be helpful. Thank you and goodbye.

In Exercises 15–22, test the claim about the population variance or standard deviation at the level of significance Assume the population is normally distributed.
Claim: σ^2>19, α=0.1. Sample statistics: s^2=28, n=17

Key Concepts

Population Variance and Standard Deviation

Hypothesis Testing

Level of Significance (α)

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