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.

ACT Mathematics and Science Scores The mean ACT mathematics score for 60 high school students is 20.2. Assume the population standard deviation is 5.7. The mean ACT science score for 75 high school students is 20.6. Assume the population standard deviation is 5.9. At α=0.01, can you reject the claim that ACT mathematics and science scores are equal? (Source: ACT, Inc.)

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. A health researcher wants to compare daily water consumption between two groups of adults. For a sample of 50 office workers, the mean daily water intake is 1.8 L. Assume the population standard deviation is 0.5 L. For a sample of 60 outdoor workers, the mean daily water intake is 2.0 L, with a population standard deviation of 0.6 L. At the alpha equals 0.01 level of significance, can you reject the claim that the average water consumption is the same for office workers and outdoor workers? Awesome. So it appears for this particular prompt, we're given a level of significance alpha of 0.01, and we're asked if we can reject the claim that the average water consumption is the same for office workers and outdoor workers. So, are we able to reject this particular claim? Yes or no? And that is our final answer we're ultimately trying to solve for. So now that we know what we're trying to solve for, let's read off our multiple choice answers to see what our final answer might be. A is, yes, the test statistic falls in the rejection region, so we can reject the claim that mean water consumptions are same. B is no, there is insufficient evidence to conclude that the average daily water consumption differs between office workers and outdoor workers. C is no. The test statistic is greater than the critical value, so we fail to reject the claim. And D is insufficient data. Awesome. So our first step that we need to take in order to solve this problem is we need to state our hypotheses. So we need to note what our null and alternative hypotheses are. So starting with our null hypothesis first, which will denote as H0. As we should recall, we'll be able to, using the information provided to us, we could state that the null hypothesis is mu1 is equal to mu 2, where mu denotes the mean. So mu1 is equal to mu2, meaning that there's no difference in means. And as for our alternative hypothesis, which will denote as HA, our alternative hypothesis on the other hand, we should recall that we're able to write that mu1 cannot equal mu2 because the means are different. And we can say this because this is a two-tailed Z test for this particular prompt. So we need to be able to recognize that this is a two-tailed Z test. So with that in mind, our second step now is we need to determine the value for our test statistic. So we need to recall and use the equation to solve for Z, our test statistic, which Z is equal to X 1 minus X 2 divided by the square root of sigma 12 divided by N1 plus. Sigma 22 divided by N2. So now we need to plug in all of our known variables to solve for the numerical value of Z, so we will find that this is equal to 1.8 minus 2.0 divided by the square root of 0.5 to 2 divided by. 50 plus 0.6 to the power 2 divided by 60. So when we plug in this expression into our calculator and we round to three decimal places, you will find that it is approximately equal to -1.907. Fantastic. So now, moving right along here, our 3rd step now is we need to determine the critical values and the rejection region. So let us recall once again that we're told that we have a level of significance of alpha equals 0.01. And we're trying to find the critical Z values for a two-tail test. So this is a two-tail test which we'll call TTT for short. So that will mean that Z subscript alpha divided by 2 will be equal to + or minus 2.576, and that will mean that the rejection region will mean that we can reject the null hypothesis if Z is less than -2.576 or when Z is greater than 2.576. So let's say that again. So, we can safely reject the null hypothesis if Z is less than -2.576 or Z is greater than 2.576. Awesome. So now for our fourth step, we need to make our final decision. So we're told once again as we should recall that Z is approximately equal to -1.907. So we can go ahead and state that since -1.907 is not in the rejection region, we fail to reject the null hypothesis, so I'll put an X next to H0 where H0 denotes the null hypothesis. So that means once again because -1.907 is not within our rejection region, we fail to reject the Newell hypothesis, and because that is the case, we can make our final conclusions as such. At the alpha equals 0.01 level of significance, there is insufficient evidence to conclude that the average daily water consumption differs between office workers and outdoor workers, meaning that without a doubt, our final answer. Has to be multiple choice answer letter B, which once again is no. There is insufficient evidence to conclude that the average daily water consumption differs between office workers and outdoor workers. 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)

Watch next