The data set represents the number of minutes a sample of 27 p...

Question

The data set represents the number of minutes a sample of 27 people exercise each week.

108 139 120 123 120 132 123 131 131

157 150 124 111 101 135 119 116 117

127 128 139 119 118 114 127 142 130

a. Construct a frequency distribution for the data set using five classes. Include class limits, midpoints, boundaries, frequencies, relative frequencies, and cumulative frequencies.

Accepted Answer

Hi everyone. Let's take a look at this practice problem. This problem says the data set below shows the weekly study times in minutes of a sample of 27 people. Instruct a frequency distribution for the data set using 5 classes, including class limits, midpoints, boundaries, frequencies, relative frequencies, and cumulative frequencies. And we're given the data of 108, 139, 120. 123, 120, 132, 123, 131, 131, 157, 150, 124, 111, 101, 135, 119, 116, 117, 127, 128, 139, 119, 118, 114, 127, 142, and 130. So, we need to construct a frequency distribution. So the first thing we need to do is determine the range of our data set. And recall that the range of our data set is going to be equal to our maximum value minus our minimum value. So, here, if we look at our data set, we see that we have a maximum value of 157, and a minimum value of 101. That means our range is going to be equal to 157 minus 101, which is going to be equal to 56. We're gonna use that range to actually calculate our class width. So this is going to be the width of each of our class limits. And so recall that our class width is equal to our class, or sorry, is equal to our range, which is 56. Divided by the number of classes, which in this case we're told the problem that we need 5 classes, well, 56 divided by 5, which is going to be equal to 11.2. But we need to round this up to the next whole number, so that means that we're gonna round this up. So that our class width is equal to 12. So we're gonna have to use this class width to struct our class limits. And so we're just going to look at our first class limit and do the calculations for the midpoint boundaries frequencies relative frequencies for this. And as an example, and then use a table to show the remaining values. So, we're gonna be looking at the class limit for our first class limit. We're going to have our minimum value, which is 101, and then we'll have to have 1212 different values in this limit. So we'll have 101 to 112. So that will give us 12 different values to look at. So next we need to determine the midpoint, and recall that the midpoint. Is the point that is in the middle of this interval. So, recall that our midpoint is going to be equal to our upper limit, which is going to be 112, plus our lower limit, which is 101, and then we want to take that sum and divide it by 2. So that means the midpoint for our first class limit is going to be equal to. 106.5. Next, we need to determine the boundaries. And recall that the boundaries. are given by For our lower boundary, we're going to have our lower limit of 101, and we're gonna subtract 0.5 from it. And then for our upper bound. We're going to have our upper limit, which is 112, and we're gonna add 0.5 to it. So, for our boundaries, in this case, it's going to be 100.5. 2 112.5. Next, we need to determine the frequency. And recall that the frequency is just how many values in our data set fall within our class limits. And here we actually see that we have 3 values. We have 108. 111 and 101. So that means that the frequency for this case is going to be equal to 3. Next, we need to calculate our relative frequency. And recall that the relative frequency is going to be equal to our frequency, which in this case is 3. Divided by the total number of data points, which in this case is 27. So we'll have 3 divided by 27, which when we evaluate that expression is going to be equal to 1 divided by 9, which is 0.11. So we're gonna need to repeat these calculations for each of our 5 classes. And so this is actually best um summarized in a table. So, on the screen here, we have a table, and in the first column, we have our class limits, and we have the values of 101 to 112, 113 to 124, 125 to 136, 137 to 148, and 149 to 1160. In the next column, we have the midpoint. And when we calculate this for each of our classes, we get the values of 106.5, 118.5, 130.5, 142.5, 154.5. The next column are the boundaries, and calculating this for each of our classes. We get the values of 100.5 to 112.5, 112.5 to 124.5, 124.5 to 136.5, 136.5 to 148.5, and 148.5 to 160.5. The next column is um the frequency, and calculating the frequency for each of our classes, we get the values of 3, 11, 83, and 2. The next column is the relative frequency, and calculating this again for each of our classes, we get the values of 0.11, 0.41, 0.30, 0.11, and 0.07. And in the final column, we have the cumulative frequency. I recall that the cumulative frequency is just a writing sum of the frequencies up to this point. So, in adding up the frequencies up to this point for each of our classes, we get the values of 3, 1422, 25, and 27. And so the values that are listed in this table here are going to be the answer to this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

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Key Concepts

Frequency Distribution

Class Limits and Midpoints

Relative and Cumulative Frequencies

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