Constructing Confidence Intervals for μ1-μ2. You can construct...

Question

Constructing Confidence Intervals for μ1-μ2. You can construct a confidence interval for the difference between two population means μ1-μ2 , as shown below, when both population standard deviations are known, and either both populations are normally distributed or both n1>= 30 and n2>=30 . Also, the samples must be randomly selected and independent.

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In Exercises 29 and 30, construct the indicated confidence interval for μ1-μ2 .

Software Engineer Salaries Construct a 95% confidence interval for the difference between the mean annual salaries of entry level software engineers in Santa Clara, California, and Greenwich, CT, using the data from Exercise 27.

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 nutritionist compares the average daily caloric intake of two groups. Group 1 athletes and group 2 non-athletes. For group 1 N1 is equal to 36, X 1 is equal to 2850. Sigma 1 is equal to 320. For group 2, N2 is equal to 42, X 2 is equal to 2600, Sigma 2 is equal to 280. Construct a 99% confidence interval for the difference in means mu 1 minus m2. What is the interval? Awesome. So it appears for this particular problem we're asked to do two things. We're asked to, firstly, construct a 99% confidence interval for the difference in means for mu 1 minus m2, and we're ultimately trying to solve for what our final answer that we're trying to solve for is we're trying to determine what is the interval of this particular. set. So we're trying to determine what the interval is, and that is our final answer we're ultimately trying to solve for. So now that we know that we're trying to solve for the interval, let's read off our multiple choice answers to see what our final answer might be, noting that they're all written in parentheses. So A is 50.21449.79. B is 124.37475.63. C is 73.18426.82. And finally, D is 110.92,489.08. Awesome. So with that in mind, our first step that we need to take in order to solve this problem is we need to identify and write down all of our known data. So we're told that X 1 is equal to 2850. We're also told that sigma 1 is equal to 320. And we're also told that N1 is equal to 36. We're also told that X 2 is equal to 2600. We're also told that sigma 2 is equal to 280, and we're also told that N2 is equal to 42. Fantastic. So now, with that in mind, our second step now is we need to recall and use the following formula, which states that parentheses X 1 minus X2. Plus or minus Z subscript 0.005. Multiplied by the square root of sigma 1 square divided by N1. Plus, sigma 2 square divided by N2. Fantastic. So now, Our third step that we need to take is we need to note that for a 99% confidence, that will mean that Z subscript 0.005 will be equal to 2.576, fantastic. So now, our fourth step, and let's make a little note here that it will equal 2.576, and this is because we have a 99% confidence interval, which I'll call CI for short. So moving right along here. So now for our fourth step, we need to determine the difference in means, meaning we need to take 2850 minus 2600, which will be equal to 250. So now our 5th step is we need to determine the standard error, which will denote as SE. So let us recall and use the equation to solve for the standard error. So the standard error SE is equal to the square root of sigma 12 divided by N1 plus sigma +22 divided by N2. Fantastic. So when we plug in our known variables to solve for our standard error, we will find that it's equal to the square root of 320 squad divided by 36 plus 280 square divided by. 42. And when we plug that into our calculator and we round to two decimal places, we will find that it's approximately equal to 68.64. Fantastic. So now, our sixth step is we need to determine the margin of error, which will denote as ME for short, and the margin of error is equal to 2.576 multiplied by 68.64. Awesome. And when we plug that into our calculator, we will find when we round to three decimal places that it's approximately equal to 176.817. Fantastic. So now, moving right along here. Now we need to determine our confidence interval. So that's our 7th step. So let us note that we're, since we're so for our confidence interval, we need to focus our attention first on the lower limit, which we'll call LL for short, and for the lower limit, we need to take 250 minus 176.817, which is equal to 73.183. And for our upper limit, which I'll call UL for upper limit, we need to take 250 plus 176.817, which will be equal to 426, so 426. 817. Fantastic. So now, now we've determined our lower limit and our upper limit. Our eighth and final step is we need to round to two decimal places and put our final answer in parentheses to write our interval, and we will find that it's parentheses. Our final answer will be parentheses 73.18, so 73.18. 426.82. And that's it. That is our final answer. Hooray, we did it. So looking at our multiple choice answers, our final answer, without a doubt, will have to be multiple choice answer letter C, which once again is parentheses 73.18,426.82. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Confidence Interval

Independent Samples

Normal Distribution and Sample Size

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