[DATA] Carpoolers The following data represent the percentage ...

Question

[DATA] Carpoolers The following data represent the percentage of workers who carpool to work for the 50 states plus Washington, D.C. Note: The minimum observation of 7.2% corresponds to Maine and the maximum observation of 16.4% corresponds to Hawaii.
a. Find the five-number summary. b. Construct a boxplot. c. Comment on the shape of the distribution.

a. Find the five-number summary. b. Construct a boxplot. c. Comment on the shape of the distribution.

Table displaying carpooling percentages for 50 states plus D.C., ranging from 7.2% to 16.4%, arranged in rows and columns.

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says, listed below are the self-reported weights in kilograms of adults in the corresponding measured weights based on data from a national health survey. Well, the texts we're given a table that has two columns. In the first column, we have the self-reported weight in kilograms, and in this column, we have the values of 67.5, 70.2, 62.8, 69.5, 70.8, 59.5, 66.2, 63.5, 50. 5.0, 64.5, 67.0, and 71.5. In the second column, it is labeled measured weight in kilograms, and in this column we have the values of 66.8, 69.5, 65.2, 68.0, 71.2, 61.0, 65.5, 66.5, 56.8, 73.8, 66.2, and 70.2. Below the table we're getting an empty graph. And we're asked to compute the differences in the self-reported weight minus the measured weight. And to use these differences to construct a box plot and include the values of the five number summary. So, the first thing we need to do is to actually calculate the differences in the two weights. And this is actually best done in our table, and so we'll just add a third column to our table and label that column differences. And so this is going to be the self-reported weight minus the measured weight. And when we calculate this for each of our pairs of weights, we get the following values. 0.7, 0.7, -2.4, 1.5, minus 0.4, minus 1.5, 0.7, minus 3.0, minus 1.8, minus 9.3, 0.8, and 1.3. Now, um, this is going to be the answer for the first half of the problem. Now, for the second half of our problem, we need to create a box plot and to determine the values of the 5 number summary. The first thing we need to really do is to order our differences in ascending order, so starting from the smallest and going to the largest. In doing so, we will get the sequence of minus 9.3, minus 3.0. Minus 2.4. -1.8. -1.5. -0.4. Then 0.7, then 0.7 again, and then 0.7 for a third time. 0.8 1.3 And 1.5. And so, we can actually use the sequence to read off a couple of the numbers for our 5 number summary. So, the first value that we can just read off is going to be our minimum value. And so that is going to be the smallest value in our sequence, which is going to be equal to -9.3. We can also read off the maximum value, which is the largest number in our sequence, which is going to be 1.5. So those are two of the 5 number summaries that we were looking for. We also will need the median, and so recall that that is going to be the value in the middle of our number sequence. And here we have an even number of values. So we're actually be looking at the midpoint between the two values in the middle. So that's gonna be the midpoint. Of minus 0.4 and 0.7. So we're called to calculate the midpoint, we need to add our two values together, so we'll have minus 0.4 plus 0.7. And then we'll take that quantity and divide it by 2. In doing so, that means that our median is going to be equal to 0.15. So that is our next value in our five number summary. The next two values that we need to calculate are going to be our 1st quartile and 3rd quartile. So, for our first quartile, which will leave us Q1, That is going to be equal to the median of the first half of our values. So, it's going to be the median of -9.3, 3.0, minus 2.4, minus 1.8, 1.5, and minus 0.4. And again, we have an even number of values here. So, we're gonna be looking at the midpoint between the two values in the middle. So we're gonna be looking at the midpoint between -2.4 and -1.8. And so, for Q1, that's going to be equal to the midpoint, which is going to be equal to minus 2.4 plus minus 1.8, and that quantity is going to be divided by 2. And when we calculate this, that means that Q1 is going to be equal to. -2.1. Next, we'll need to find our third quartile, Q3, and recall that this is going to be the median of the second half of our data. So it's going to be the median of 0.7, 0.7, 0.7, 0.8, 1.3, and 1.5. Again, we have an even number of values here. So we're going to be looking at the two values in the middle of our sequence here, and so that's going to be The 0.7 and 0.8, and so we need to find the midpoint between those two points. And that means Q3 is going to be equal to the quantity of 0.7, plus 0.8 in quantity, divided by 2. Which that means that Q3 is going to be equal to 0.75. So those are the last two values that we needed for our 5 number summary. So just to recap for our 5 number summary, we have that the minimum is -9.3, the maximum is 1.5, the median is 0.15, Q1 is minus 2.1, and Q3 is 0.75. Now, before we can use these values to create our box plot, we first need to determine if there are any outliers for our values here. And for that, we'll need to first calculate the interquartile range, which we'll label as IQR. And we call that our inner quartile range is going to be equal to Q3 minus Q1. Which is going to be equal to 0.75. Minus 2.1, which is going to be equal to 2.85. So now we can use our inner quartile range here to calculate our upper bound and lower bounds. So for our lower bound, which we'll label as LB, recall that that is going to be equal to 1. -1.5, multiplied by an end to a quartile range IQR. And so, substituting in our values, this is going to be equal to minus 2.1. -1.5, multiplied by 2.85. And when we evaluate this expression, that means our lower bound is going to be equal to -6.375. Now, for our upper bound, UB, this is going to be equal to, if you recall, U3 plus 1.5 multiplied binary interquartile range IQR. So substituting in values, this is going to be equal to 0.75 plus 1.5, multiplied by 2.85. And when we evaluate this expression, this is going to be equal to 5.025. So, we need to determine if any of our values fall outside of these bounds. And we actually only have one that falls outside of these bounds, and that's going to be our minus 9.3. That is our only outlier in our sequence. So now we have enough information to draw our box plot. And so, the first thing we need to do is actually draw our box. And so we call that our box, goes from the first quartile to our third quartile. So, we're going to draw a vertical line at Q1, which is minus 2.1. And then we'll draw another vertical line on our graph. At Q3, which is 0.75. And then we'll connect the tops of those two vertical lines together to form the top of our box. And then we'll connect the bottom of those two vertical lines to create the bottom of our box. Now, inside of a box, we'll need to draw an additional vertical line to indicate our median, and our median had a value of 0.15. We'll draw a vertical line at 0.15 on our graph. So now we need to draw our whiskers. Now, for the whisker on the right, it's gonna go from the largest value. That is not an outlier. And so in our case, that is going to be 1.5, so we'll draw a short vertical line with the whisker at 1.5, and then connect that vertical line. With a horizontal line to the vertical line for Q3. Now, for the whisker on the left, we're going to have it at the smallest non-outline value, which in our case is going to be equal to -3. So, we'll draw a short vertical line at -3, and then connect that with a horizontal line. 2 are vertical line 4Q1. Now, the final thing that we need in our graph here is to identify our outlier. And since we only had one and it was at a value of -9.3, we're going to place a dot at -9.3 on our graph. So, that will indicate our outlier. And so the graph that we have on the screen here is going to be the graph that this question was asking for. So, I hope that this explanation has been useful, and I'll see you in the next video.

Key Concepts

Five-Number Summary

Boxplot Construction

Distribution Shape and Skewness

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