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<40, α=0.01 . Sample statistics: s=40.8, n=12

Accepted Answer

Hello everyone. Let's take a look at this question together. A manufacturer claims that the standard deviation of the weights of its cereal boxes is less than 25 g. At the alpha equals 0.01 significance level, test this claim using the following sample data sample standard deviation S equals 26.2 g, sample size N equals 15. Assume the weights are normally distributed. Is it answer choice A, there is no sufficient evidence to support the claim that the population standard deviation is less than 25 at the alpha equals 0.01 significance level. Answer choice B, there is sufficient evidence to support the claim that the population standard deviation is less than 25 at the alpha equals 0.01 significance level, or answer choice C, there is not enough information. So in order to solve this question, we have to test the claim by the manufacturer that the standard deviation of the weights of their cereal boxes is less than 25 g at the alpha equals 0.01 significance level, and we know from the information provided, the sample standard deviation is 26.2 g and the sample size N equals 15, as well as we must assume the weights are normally. So the first step in solving this problem and testing our claim is to state the hypotheses, starting with our null hypothesis, which is the standard deviation equals 25. And then our alternative hypothesis is that the standard deviation is less than 25. And then next we compute the test statistic where we use the chi square test statistic for standard deviation. Which is given by the following formula of chi square is equal to and minus 1 in parentheses multiplied by the sample variants divided by the population variants. And plugging in the values that we are given into our chi square test statistic formula, the value for the chi square test statistic is 15.3763, and now we can determine the critical values where we need to find the critical value for alpha equals 0.01, a left tail, and we know our degrees of freedom is equal to 14, and from the chi square distribution tables. Our chi square critical value is equal to 4.660. And then using the chi square critical value, we compare the test statistic to that critical value. Which we note that 15.3763 is greater than 4.660. Therefore, the test statistic is not in the critical region, and so we can conclude that we reject our null hypothesis if the chi square test statistic is less than 4.660. However, we know that in this case. Failed to reject the null hypothesis as our chi square test statistic is greater than the critical value of 4.660, so we failed to reject the null hypothesis as there is not enough evidence to support the claim that the population standard deviation is less than 25 at an alpha equals 0.01 significance level. Therefore, the correct answer choice is answer choice A, as there is no sufficient evidence to support the manufacturer's claim that the standard deviation of the weights of cereal boxes is less than 25 g at an alpha equals 0.01 significance level, 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<40, α=0.01 . Sample statistics: s=40.8, n=12

Key Concepts

Hypothesis Testing

Significance Level (α)

Chi-Square Test for Variance

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