Shifting Data Sample annual salaries (in thou...

Question

Shifting Data Sample annual salaries (in thousands of dollars) for employees at a company are listed.

40 35 49 53 38 39 40

37 49 34 38 43 47 35

c. Each employee in the sample takes a pay cut of $2000 from their original salary. Find the sample mean and the sample standard deviation for the revised data set.

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says a company's quarterly bonuses in thousands of dollars for a group of employees are as follows 22, 25, 28, 31, 24, 27, 29, 23, 30, 26, 25, and 28. If each employee receives a reduction of $3000 in their bonuses, what are the sample mean a sample standard deviation of the adjusted data set? So the first thing we need to do is to adjust our data set. So, we're told that each employee receives a reduction of $3000. And since the values in our data set are given in thousands of dollars, that means we'll just have a reduction of 3 from the values that we were given. So, doing that reduction, we will have the data set of 19. 22 25 28 21 20 4 26 2027. 2322. And 25. So we need to calculate the sample mean and sample standard deviation for this new data set. So recall that your sample mean X bar is given by the formula that it is equal to the sum. Of X of I divided by N, where X of I is an individual data point in our data set, and N is the total number of data points that we have, which in this case is 12. So using this formula, that means that X bar is going to be equal to, and the numerator will have the quantity of 19 + 22 + 25 plus 28 + 21, plus 24. Plus 26 + 20 + 27 plus 23 plus 22. Plus 25 in quantity, and all of that is going to be divided by the total number of data points that we have, which in this case is 12. And when we evaluate this expression, this is going to be equal to 23.5. Which means the sample mean, if we're looking at it in dollars instead of thousands of dollars, is going to be. 23,500 So that's gonna be our answer for the first part of this problem. Now, for the second part, we need to calculate the sample standard deviation. And so, recall that our sample standard deviation S is equal to the square root of the quantity of the sum. Of the quantity of X of I minus X bar in quantity squared divided by the quantity of N minus 1 in quantity. So, using this formula, that means that S is going to be equal to. The quantity, and here we're just going to write things a little different. So, we're going to write the square root as raising the quantity to the 1 half power. And so that means we're going to also factor out the 1 divided by the quantity of N minus 1. So we'll have 1 divided by the quantity of 12 minus 1. In quantity, and that gets multiplied by the quantity, and here we'll have the sum of X by minus X bar in quantity squared. And doing so, we'll have the quantity of 19 minus 23.5 in quantity squared. Plus the quantity of 22 minus 23.5 in quantity squared, plus the quantity of 25 minus 23.5 in quantity squared, plus the quantity of 28 minus 23.5 in quantity squared, plus the quantity of 21 minus 23.5. In quantity squared. Plus The quantity of 24 minus 23.5 in quantity squared, plus the quantity of 26 minus. 23.5. In quantity squared Plus The quantity of 20. -23.5. In quantity squared plus the quantity of 27 minus 23.5 in quantity squared, plus the quantity of 23 minus 23.5 in quantity squared, plus the quantity of 22 minus 23.5. In quantity squared, and finally plus the quantity of 25 minus 23.5. In quanti squared. Then we'll have an in quantity, then another in quantity, and that gets raised to the one half power. So again, as I said before, we're rewriting the square root as raising the quantity to the one half power. And so when we evaluate this expression. We get that S is approximately equal to. 2.81. And so we actually want this in dollars, instead of thousands of dollars, that means that S is going to be approximately equal to. 2000 $810. That is going to be our answer for the second half of the problem. So just to recap, we have that X bar is equal to $23,500 and S is equal to $2810. So that's going to be our final answer for this problem. So I hope that this explanation has been useful and I'll see you in the next video.

Key Concepts

Sample Mean

Sample Standard Deviation

Uniform Transformation

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