Graphical Analysis In Exercises 21–24, you ar...

Question

Graphical Analysis In Exercises 21–24, you are asked to compare three data sets.

(c) Estimate the sample standard deviations. Then determine how close each of your estimates is by finding the sample standard deviations.

i. Bar graph showing frequency distribution of data entries ranging from 4 to 10, with peaks at 6 and 7.

ii. Bar graph showing frequency distribution of data entries from 4 to 10, with peaks at 4, 5, 9, and 10.

iii. Bar graph showing frequency distribution with data entries ranging from 4 to 10, highlighting values around 6 to 8.

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, we're going to read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Use the graph below to estimate the sample standard deviation using midpoints and frequencies. And calculate the actual standard deviation using individual data values. Compare both results and select the appropriate option from below. Awesome. So it appears for this particular problem we're asked to use our graph, which appears to be a bar graph, and we're asked to use our graph to help us estimate the sample standard deviation using midpoints and frequencies to help us calculate the actual standard deviation using individual data value. So. Once again, we're trying to estimate the sample standard deviation using midpoints and frequencies, and we're asked to calculate the actual standard deviation using individual data values. And then finally we're asked to compare both results and select the appropriate option from our multiple choice answers. So with that in mind, let's quickly look at our graph before we look at our multiple choice answers. So our Y axis, the vertical axis, denotes the number of students. Our horizontal X axis represents the hours of study. So it appears for our hours of study, we have 45678, and 9 hours, and the number of students that study 4 hours is 2, the number of 5 hours studied is 4, the number of 6 hour study is 5, the number of 7 hours studied is 6, the number of 8 hours studied is 5, and then finally 9 hours studied is 3. Fantastic for 3 for the number of students that studied for 9 hours. So now that we have a good visualization of what's happening for this particular prom, let's read off our multiple choice answers to see what our final answer might be. So A is the estimated standard deviation 1.64 is slightly higher than the actual standard deviation 1.49. B is the estimated standard deviation 1.29 is lower than the actual standard deviation 1.49. C is both estimated and actual standard deviations are the same 1.49, and finally, D is both estimated and actual standard deviations are the same 1.64. Awesome. So, first off, our first step is we need to extract the data from the graph and then create a table to help us organize our data. So with that in mind, once again, when we extract the data from our bar graph, We will be able to create the following table, and as you can see our far left column is for the hours study or the hours of study, which is our values for X, which is for 45678, and 9, and the frequency of our hours in order going from 4 to 9 are 24565, and 3. Awesome. And our second step is we need to, now that we've created our table to extract all of our data that we pulled from our graph, now our 2nd step. Once again, we need to estimate the sample mean and the sample standard deviation. So with that in mind, we could create the following table to help us organize our data because we're trying to determine. The estimated value for the sample mean and the sampled standard deviation. So we could create the following table. As you can see, going from our far left column working our way to the far right column, we have our value for X subscript I, so our hours for 4 to 9, and then we have our frequency, which is F subscript I, which is As a total of 25, so that's when we add 2 + 4 + 5 + 6 + 5 + 3. Then we have our column for XI multiplied by FI, which is 8, 2030, 42, 40, 27, which has a total of 167 when you add all those values together. Then for our value of Xi minus X bar, we have -2.68, 1.68, 0.68, 0.32, 1.32, 2.32. And then finally for our next column we have parenthesis XI minus X to the power of 2, which gives us 7.1824, 2.8224. 0.4624 0.1024 1.7424 and finally 5.3824. And then finally for our last column on the far right, we have FI multiplied by parentheses Xi minus X to the power of 2. Which gives us 14.3648, 11.2896, 2.312, 0.6144, 8.712, and 16.1472, which gives us a total of when you add those values together, 53.44. Awesome. So as we should recall, we should note that N is equal to the sum of F, where F once again is the frequency. And the sample mean, which is X bar, is going to be equal to the sum. Of FI, so the sum of F I multiplied by XI. All divided by N, which is going to be equal to me plug in or known values, 167 divided by 25, which is gonna be equal to when we round to two decimal places. 6.68. Fantastic. And for the formula, as we should recall, the solve for the sample standard deviation, which we'll call SEST, so the this is the estimated standard deviation. So the estimated standard deviation is equal to the square root of The sum of F of Xi minus X to the 2 divided by N minus 1, which is going to be equal to the square root of 53.44 divided by 25 minus 1. And when we plug that into our calculator, we will find that this is approximately equal to 1.49. And that is one of our answers. Hooray, we did it. So now, moving right along here, our 3rd step As we need to calculate the actual standard deviation using our raw data. So for this particular case, our sample mean, which is X bar, is gonna be equal to 4 + 4 + 5 + 5 + 5 + 5. So there's 45, so 1234. Plus 6 + 6 + 6 + 6 + 6 + 7 + 7 plus 7. Plus 7 + 7, so we need to make sure, so we have +12345. We have 5 6s, so +12345. Then we have +123456. We have +67, so +12345, so we need to add one more 7. Then we need to add +12345, we need to add +58, so 1. 23. So 12345, so 5 8s, and then we need to add 3 9s, so plus 9, plus 9, plus 9. Fantastic. Then we need to divide everything by 25, and when we do that, we will find that X bar is equal to 6.68. Fantastic. We also need to recall and use our formula for the standard deviation. So S or standard deviation is equal to the square root of the sum of X minus X to the power 2 divided by N minus 1. So that will mean. That we could go ahead and plug in our known data points, so we could go ahead and write. That The sum Of X minus X 2 is going to be equal to 4 minus 6.68 2 multiplied by 2 plus 5 minus 6.68 2 multiplied by 4 + 6 minus 6.68 to the power of 2 multiplied by 5 + 7 minus 6.68 to the power 2 multiplied by 6. Plus parentheses 8 minus 6.68 to the power of 2 multiplied by 5 + 9 minus 6.68. To the power of 2 multiplied by 3. Which will give us an answer of when you perform that calculation in your calculator, it will be equal to 53.44. Fantastic. So solving for our standard deviation, we will find that S is equal to the square root of 53.44 divided by 25 minus 1. So once again, the square root of 53.44 divided by 25 minus 1. Which when we plug it into our calculator, we will find when we round to two decial places that it's approximately equal to 1.49, and that is our second answer for the actual standard deviation. Hooray, we did it. So now let's make our final comparison and make our final conclusions. So our estimated standard deviation is approximately equal to 1.49, and our actual standard deviation is also approximately equal to 1.49. So therefore, in conclusion, the estimate using grouped frequencies and midpoints is the same as the actual value using our raw data. And this is the case because the data points are discrete and directly match the frequency distribution used in our estimation. So that will mean, without a doubt that our final answer will have to be multiple choice answer letter C, which once again is both estimated and actual standard deviations are the same 1.49. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Graphical Analysis In Exercises 21–24, you are asked to compare three data sets.

(c) Estimate the sample standard deviations. Then determine how close each of your estimates is by finding the sample standard deviations.

i.
ii.
iii.

Key Concepts

Sample Standard Deviation

Frequency Distribution

Estimation Techniques

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