Testing a Difference Other Than Zero Sometimes a researcher is...

Question

Testing a Difference Other Than Zero Sometimes a researcher is interested in testing a difference in means other than zero. In Exercises 27 and 28, you will test the difference between two means using a null hypothesis of Ho: μ1-μ2=k, Ho: μ1-μ2>=k or Ho: μ1-μ2<=k . The standardized test statistic is still

Software Engineer Salaries Is the difference between the mean annual salaries of entry level software engineers in Santa Clara, California, and Greenwich, Connecticut, more than $4000? To decide, you select a random sample of entry level software engineers from each city. The results of each survey are shown in the figure at the left. Assume the population standard deviations are σ1=$14,060 and σ2=$13,050 . At α=0.05, what should you conclude? (Adapted from Salary.com)

Entry level software engineer salaries in Santa Clara, CA, $88,900, and Greenwich, CT, $81,600, with sample sizes noted.

Accepted Answer

Hello everyone. Let's take a look at this question together. A study compares the mean monthly rent for two cities, City X and City Y, and wants to know if the difference is less than $1200. For City X, we have a sample mean of $2350 a sample size of 22, and a population standard deviation of 410. Then for city Y, we have a sample mean of 1250, a sample size of 20, and a population standard deviation of 390. Test the null hypothesis that mean one subtracted by mean 2 is equal to $1200 versus our alternative hypothesis, which is that mean one, subtracted by mean 2 is less than $1200 at alpha equals 0.05. What is the result? Is it answer choice A, reject the null hypothesis? Answer choice B, fail to reject the null hypothesis, answer choice C, test statistic is 0, or answer choice D, test statistic is undefined. So in order to solve this question, we have to recall how to test a null hypothesis versus an alternative hypothesis at alpha equals 0.05 to determine what is the result for this study comparing the mean monthly rent for the two cities, which are CX and C Y. And so the first step in determining what is the result of this hypothesis test, we must state the hypotheses. Which from the given information, our null hypothesis is that population mean one, subtracted by population mean two is equal to 1200. And our alternative hypothesis is that our population mean one, subtracted by population mean 2 is less than 1200, and then we can identify the sample means, the sizes, and the standard deviations, which from the given information, we note that City X has a sample mean of 2000. 350, a sample size of 22, and a population standard deviation of 410, and City Y has a sample mean of 1250, a sample size of 20, and a population standard deviation of 390. And now we can compute the test statistic. Using the formula Z is equal to sample mean 1 subtracted by sample mean 2, subtracted by 1200 divided by the square root of population variance 1 divided by sample size 1 plus population variance 2 divided by sample size 2. Starting with our first calculation of sample mean 1 minus sample mean 2, we have 2350 subtracted by 1250, which is equal to 1100. Then we have population variance one divided by sample size 1, which is 410 square divided by 22, or approximately 7,640.91, and then population variance 2 divided by sample size 2, we have 390 squared divided by 20, which is approximately 7,605. So plugging those values into the square root. We have the square root of 7,640.91 plus 7,605, which is equal to 123.47. So substituting those values into the formula for the test statistic, our test statistic Z is approximately 0.810. And then we can determine the critical value for alpha equals 0.05, which this is a left tailed test, so our critical value for Z is equal to -1.645. And since our calculated test statistic of -0.810 is greater than -1.645, we fail to reject the null hypothesis. Therefore, looking at our answer choices, we can identify that answer choice B is the correct answer. As we know, testing this hypothesis, we failed to reject the null hypothesis, since our calculated test statistic is greater than the rejection region. So answer choice B is the correct answer, and all other answer choices are incorrect. I hope you found this video to be helpful. Thank you and goodbye.

Key Concepts

Hypothesis Testing

Standardized Test Statistic

Significance Level (α)

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