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. Scatter plot showing data points distributed around the values 10 to 14, indicating variability in the data sets.

ii. A scatter plot showing data points distributed around the values 10 to 14, with a peak at 12.

iii. A dot plot showing data points clustered around the value 12, with values ranging from 10 to 14 on the horizontal axis.

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. Use the dot plot to estimate the sample standard deviation using midpoints and frequencies, then calculate the actual standard deviation using individual data values. Compare both results and select the appropriate option from below. Awesome. So we have our dot plot that is given to us, and it starts at the value of 0 and it goes 01234, and 5. And it looks like we have a frequency of 2 for 0, a frequency of 3 for 1, a frequency of 4 for 2, a frequency of 3 for 3, a frequency of 2 for 4, and a frequency of 1 for 5. Awesome. So now that we have a visualization of what's happening for this particular prompt, let us read off our multiple choice answers to see what our final answer might be, noting once and again. That we're ultimately trying to determine the sample standard deviation using the midpoints and frequencies, and then 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, A is the estimated and actual standard deviations are both about 1.47, so the estimate is very close. The estimated standard deviation is much higher than the actual value showing the estimate is not reliable for B. And then C is the estimated standard deviation is much lower than the actual value. And finally, D is the actual standard deviation is about 2.4, and the estimate is about 1.0, so the estimate is not accurate. Awesome. So, with that in mind, our first step in order to solve this particular problem, we need to extract all the data from the data plot, so we need to use our dot plot, and when we use our dot plot, we can create the following table to help organize all of our data. So in our far left column, we have our value X, and we have the value 01234, and 5. And then we had our frequency, which I mentioned before. So for 0, we had a frequency of 2, for 13, for 24, for 33, for 42, and for 51. And if we were to write our raw data in a lot, like if we were to write our raw data from our lowest value to our highest value in numerical order, you would have 0.0.111222233344.5. Awesome. So that's when we just basically take all of our data and we write it in a Altogether from our smallest value going up to our largest value, like we group all of our like terms. So, with that said, our second step now is we need to Drum roll, we need to find the sample means. That's our second step. So our second step once again is we're finding the sample means, so we need to recall a note that X bar. is equal to the sum multiplied, or rather the sum of F multiplied by X. So the sum of F multiplied by X. Divided by N. Where we need to recall a note that the sum. Uh, so we're gonna have to recall, this is important, recall that the sum of F multiplied by X is equal to 2 multiplied by 0 + 3 multiplied by 1. Plus 4 multiplied by 2. Plus And once again, it was 3 multiplied by 1 plus 4 multiplied by 2, plus 3 multiplied by 3, plus 2 multiplied by 4 plus 1 multiplied by 5, which is gonna be all equal to, when we add it up, 33. And then N, as we should recall, is equal to the sum of F, which is equal to 2 + 3 + 4 + 3 + 2 + 1, which when we add that all together, we will find that it's all equal to 15. So therefore, without a doubt, that will mean that X bar. Is going to be equal to 33 divided by 15, which is going to be equal to, when we perform that calculation, so 33 divided by 15 is equal to 2.2. Fantastic. So now, Our next step Our 3rd step is we need to estimate the standard deviations using midpoints and frequencies. Awesome. So, we need to recall and use the following formula first. So our estimated standard deviation, so we'll denote our estimated standard deviation as S subscript EST. is equal to the square root of The sum of of. X minus X bar. To the power of 2, divided by N minus 1. Awesome. And then we could create the following data table to help collect and organize our data points. And we'll be able to create the following table with our far left column being X, or value for X, our next column being for F, then our next column being X minus X bar. Which is X minus 2.2, and then our next column will be for parentheses X minus X in parenthesis to the power of 2, and then finally our column on the far right would be F of X minus X to the power of 2. So, as we know, for our values of X, we have it from 0 to 5. And then for our values of F, which is the frequency, we had 234321. So the sum is 15. So then for X minus X bar, we have negative 2.2, 1.2, negative 0.2, 0.8, 1.8, and 2.8. And then for parentheses X minus X to the power of 2, we have 4.84, 1.44. 0.04, 0.64, 3.24, 7.84. And then finally, for our final column, we have 9.68, 4.32, 0.16. 1.92, 6.28, 7.84 with a total sum of 30.20. So that means using our table to interpret our data, that would mean that the Sum of F of X minus X squared is equal to 30.20. So therefore, S. EST, so S subscript EST, so this is our estimated. Value for the, once again, this is the standard deviation, our estimated standard deviation is equal to the square root of 30.20 divided by 15 minus 1, which is going to be approximately equal to when we round to two decimal places, 1.47. Awesome. So now, our next step. So our next step now, so this is gonna be step number 4. we need to calculate the actual standard deviation using our raw data. So now we need to note that the sample means, so X bar is going to be equal to 2.2, and we also need to recall and use the equation for the actual standard deviation. So the standard deviation S. Is equal to the square root of the sum of X minus X bar to the 2 divided by N minus 1. So therefore, we could go ahead and write that the sum of X minus X 2 is equal to 0 minus 2.2 2 multiplied by 2 plus 1 minus 2.2 to 2. Multiplied by 3. Plus parentheses 2 minus 2.2 to the power 2 multiplied by 4 + 3 minus 2.2 to 2 multiplied by 3 + 4 minus 2.2 to the power 2 multiplied by 2. Plus parentheses 5 minus 2.2 to the power of 2 multiplied by 1. Which is all equal to 30.4. So that will mean that Our estimated standard deviation for this case, so SEST is equal to the square root of. 30.4 divided by 15 minus 1, which will be also approximately equal to 1.47 when we round to two decimal places. So that means that when we perform our comparison and make our final conclusions, that will mean, in which I said that this is our estimated standard deviation, technically, if you wanted to be more precise, we could call this our actual, so. Our actual standard deviation will be approximately equal to 1.47. And it's important to note that our estimate, which is SEST or estimated standard deviation, will be also approximately equal to 1.47, and that's it. We've officially solved for this problem. Hooray, we did it. So the estimate using grouped frequencies at midpoints is very close to the actual value using our raw data, and this is because the data points are discrete. And directly match the frequency distribution used in the estimation. So that means without a doubt, when we go to look at our multiple choice answers, the correct answer has to be multiple choice answer letter A. Which once again is the estimated and actual standard deviations are both about 1.47, so the estimate is very close. 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

Estimating Standard Deviation

Data Distribution

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