Testing the Difference Between Two Means In Exercises 15–24, (...

Question

Testing the Difference Between Two Means In Exercises 15–24, (a) identify the claim and state Ho and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic z, (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.

Braking Distances To compare the dry braking distances from 60 to 0 miles per hour for two makes of automobiles, a safety engineer conducts braking tests for 16 compact SUVs and 11 midsize SUVs. The mean braking distance for the compact SUVs is 131.8 feet. Assume the population standard deviation is 5.5 feet. The mean braking distance for the midsize SUVs is 132.8 feet. Assume the population standard deviation is 6.7 feet. At α=0.10 , can the engineer support the claim that the mean braking distances are different for the two categories of SUVs? (Adapted from Consumer Reports)

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. To evaluate battery life between two types of wireless headphones, a technician tests 20 units of brand A and 25 units of Brand B. Brand A has a mean battery life of 18.2. Hours with a known population standard deviation of 1.5 hours. Brand B has a mean battery life of 17.5 hours with a known population standard deviation of 1.8 hours. At the alpha equals 0.05 level of significance, can the technician conclude that there is a significant difference in the average battery life between the two brands? Awesome. So it appears for this particular problem, we're asked, given a specific level of significance of 0.05, can the technician conclude that there is significant difference in the average battery life between two brands? So that's what we're ultimately going to be trying to solve for. So now that we know what we're ultimately trying to solve for, let's take a moment to read off our multiple choice answers to see what our final answer might be. A is there is sufficient evidence to conclude that brand A has significantly higher battery life than Brand B. There is sufficient evidence to conclude that Brand B has significantly higher battery life than Brand A. C is there is insufficient evidence to conclude that the average battery life differs between Brand A and Brand B. And D is insufficient data. Awesome. So first off, in order to help us solve this problem more effectively, let's take a moment to write down all of our given and known variables. So for Brand A, which I'll call BA for short, so for Brand A, we're told that X bar 1 is equal to 18.2. We're also told that Sigma 1, sigma 1 is equal to 1.5 because that is our population standard deviation. And we're also told the value of N1, where N is our sample size, and we're told that our sample size for N1 is equal to 20. And for Brand B, which I'll call BB for short, so for brand B, Our known variables are as follows. So we're told that X 2 is equal to 17.5, sigma 2 is equal to 1.8, and N2 is equal to 25. We're also told that the level of significance alpha is equal to 0.05. We're also, we can recall and note that since we know what the population standard deviations are, we can therefore recall and use the Z test. And since we're dealing with independent variables, we can also note that we are dealing with normally distributed samples. So with that in mind, We can go ahead and start to solve for this problem. So our first step that we need to take is we need to state our hypotheses, and in order to do that, we need to focus our attention on the claim, which once again, as we should recall, the claim states that the mean battery lives are different. So focusing on our null hypothesis first, which is denoted as H0, we can say that our null hypothesis states that mu1 is equal to mu 2, or mu denotes the mean, whereas our alternative hypothesis, which I'll call HA, states that mu1 cannot equal mu 2. And we could say this because as we could see from the information given to us by the problem itself, we need to be able to recognize that this is a two-tailed situation, a two-tailed test condition. So, because of that, we can state that our null hypothesis for the mean, for mu1 cannot equal m2, and for our null hypothesis, mu 1 can equal m2. So with that in mind, Our second step now is we need to compute the test statistic value, so we need to recall and use the formula for the 2 sample Z test since we're dealing with independent samples and the population standard deviation values are. Known and given to us, so we could say that Z, as we should recall, is equal to x 1 minus X2 all divided by the square root of sigma 12 divided by N1 plus sigma +22 divided by N2. And when we plug in our known variables to solve for Z, we will find that this is equal to. 18.2 minus 17.5 divided by the square root of 1.5 square divided by 20. Plus 1.8 squared divided by 25. So when we plug this expression into our calculator, we will find that Z is approximately equal to 1.423 let me round to three decimal places. Awesome. So now our third step. we need to find the critical values and the rejection region for this particular prompt, noting that we're dealing with a level of significance of 0.05 and that this is a two-tailed test, which we'll call TTT for short. So this is a two-tailed test. So that will mean that the critical Z value, which will denote as Z subscript alpha divided by 2, will be equal to plus or minus 1.96, and that will mean that the rejection region will be Z is less than -1.96 or Z is greater than 1.96. So Z is less than -1.96 or Z is greater than 1.96. Awesome. So now, moving right along here, our fourth step now. we need to make our final decision, so we need to note that our test statistic Z is equal to 1.423. We need to note once again that our rejection region states that Z is less than -1.96 or Z is greater than 1.96, and cents -1.96. is less than 1.423, which is less than 1.96 1.96 to be precise. We fail to reject the null hypothesis, so I'll put an X next to H 0, which denotes our null hypothesis. So we fail to reject the null hypothesis, and because we reject So since we fail to reject the null hypothesis, we can make our final conclusion. Which is at the 0.05 level of significance, there is insufficient evidence to conclude that the average battery life differs between brand A and Brand B. So that will mean when we look at our multiple choice answers, the correct answer without a doubt will have to be multiple choice answer letter C. Which once again is, there is insufficient evidence to conclude that the average battery life differs between brand A and Brand B. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Hypothesis Testing

Critical Value and Rejection Region

Standardized Test Statistic (z)

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