Constructing a Confidence Interval In Exercises 31 and 32, use...

Question

Constructing a Confidence Interval In Exercises 31 and 32, use the data set to (b) find the sample standard deviation

[APPLET] Earnings The annual earnings (in dollars) of 32 randomly selected intermediate level life insurance underwriters (Adapted from Salary.com)

Table displaying annual earnings in dollars of 32 life insurance underwriters, with values organized in rows and columns.

Accepted Answer

Hi everyone. Let's take a look at this practice problem. This problem says a researcher records the monthly rent in dollars paid by 20 randomly selected graduate students, and we have the values of $1200 1350 dollars, $1250 1400, 1300, 1280, 1320, 1380, 1260, 1340, 1290, 1360, 1270, 100. 330, 1310, 1370, 1240, 1390, 1230, and 1400. Find the sample standard deviation, given that the sample mean is 1,313.5. And we're given four possible choices as our answers. For choice A, we have 11.28. For choice B, we have 19.84. For choice C, we have 1,313.50, and for choice D, we have 60.20. Now, we're asked to find the sample standard deviation of our data. So, here we call your formula for the sample standard deviation, and that's going to be S is equal to the square root of the quantity of the sum. Of the quantity of X minus X bar in quantity squared, divided by the quantity of N minus 1 in quantity. Where S here is our sample standard deviation, X is going to be one of our data points, X bar is our sample mean, and N is our sample size. Now, we're given some information in the problem. We're told that our sample mean, which is going to be X bar, is going to be equal to 1,313.5, and we're also given the sample size, which is N, and that's going to be equal to 20, since we have randomly selected 20 graduate students. So, in order to calculate our sample standard deviation, we need to calculate the sum of the quantity of X minus X bar in quantity squared, and this is actually best done using a table. So, on the screen here, we have a table that has 3 columns. In the first column, we have our data, which is labeled X. And we have the values of 1200, 1350, 1250, 1400, 1300, 1280, 1320, 1380, 1260, 1340, 1. 1290 1360, 1270, 1330, 1310, 1370, 1240, 1390, 1230, and 1400. In the next column we have the quantity of X minus X bar. So calculating this for each data point, we get the following values minus 113.5, 36.5, minus 63.5, 86.5, minus 13.5, minus 33.5, 6.5, 66.5, minus 53.5, 26.5, minus 23.5, 46.5, minus 43.5, 16.5, minus 3. 56.5 minus 73.5, 76.5 minus 83.5, and 86.5. In the final column, we'll have the quantity of X minus X bar and quantity squared. So we're just going to be squaring the previous column. In doing so, we get the following values 12,882.25, 1,332.25, 4,0032.25, 7,482.25. 182.25, 1,122.25, 42.25, 4,422.25, 2,862.25, 702.25, 552.25, 2,162.25, 1,892.25, 272.25, 12.25. 3,192.25, 5,402.25, 5,852.25, 6,972.25, and 7,482.25. So now all we need to do is to sum that final column. In doing so, we will have that the sum. Of the quantity of X minus X bar in quantity squared is going to be equal to. 68,000 855 So now we have all the information that we need to calculate our sample standard deviation. So that means that our sample standard deviation S is going to be equal to the square root. Of the quantity, and here for the sum of the quantity of X minus X bar in quantity squared, we'll substitute in the value of 68,855, and we'll divide that by the quantity of N minus 1, which is going to be 20 minus 1. And when we evaluate this expression, we see that S is going to be approximately equal to 60.20. So here we just round it to two decimal places. So this is the answer that we're looking for for this problem. And if we compare that to our answer choices, we see that the correct answer choice corresponds to answer choice D. So I hope that this explanation has been useful, and I'll see you in the next video.

Constructing a Confidence Interval In Exercises 31 and 32, use the data set to (b) find the sample standard deviation
[APPLET] Earnings The annual earnings (in dollars) of 32 randomly selected intermediate level life insurance underwriters (Adapted from Salary.com)

Key Concepts

Sample Standard Deviation

Confidence Interval

Random Sampling

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