In Exercises 9–18, construct the histograms and answer the giv...

Question

In Exercises 9–18, construct the histograms and answer the given questions.

Chicago Commute Time Use the frequency distribution from Exercise 13 in Section 2-1 to construct a histogram. Does it appear to be the graph of data from a population with a normal distribution?

Accepted Answer

Hi, everyone, let's take a look at this practice problem. This problem says a traffic study recorded the commute times in minutes of 50 individuals traveling to work in Chicago. The frequency distribution of commute times is given below, and we're given a table that has 2 columns. In the first column, we have the commute times in minutes, and it has the categories. Of 10 to 1920 to 29, 30 to 39, 40 to 49, 50 to 59, and 60 to 69. And the second column we have the frequencies, and those are 58, 1210, 9, and 6. Using the frequency distribution above, construct a histogram, then determine whether the histogram suggests that the data follow a normal distribution. So, the first thing we want to do is to create a histogram, and to do this, we're actually going to create a 3rd column here, corresponding to the midpoint of our commute times. So we're going to calculate the midpoint. That that is going to be equal to the upper bound, which we will label as UB plus the lower bound of the interval, which we will label as LB, and that quantity is going to be divided by 2. So creating a 3rd column. For the midpoint of each of our intervals of commute times. For the interval going from 10 to 19, it has a midpoint of 14.5. For 20 to 29, we have a midpoint of 24.5. For 30 to 39, we have a midpoint of 34.5. For 40 to 49, we have a midpoint of 44.5. 450 to 59, we have a midpoint of 54.5. And for 60 to 69, we have a midpoint of 64.5. So, we're actually going to use our midpoints here and our frequencies to construct our histogram. So, to construct our histogram, we can either use a spreadsheet software or manually draw it on graph paper. But in either case, on the x-axis, we'll plot the class of midpoints, and on the Y axis, we'll plot the corresponding frequencies, and we'll just need to draw vertical bars with heights equal to the frequencies for each class. Doing so, we'll get the following graph, where on the x-axis, We have the midpoint classes of 14.5, 24.5, 34.5, 44.5, 54.5, and 64.5. And we have a vertical bar drawn to the frequencies for each of those classes. So for 14.5, it has a frequency of 5, for 24.5, it has a frequency of 8. For 34.5, we have a frequency of 12, for 44.5, we have a frequency of 10, and for 54.5, we have a frequency of 9. And finally, for 64.5, we have a frequency of 6. So this here is the histogram that the problem was looking for. Next, we need to determine whether the histogram suggests that the data follow a normal distribution. So, recall that a normal distribution typically has a single peak in the center. It's symmetric on both sides of the peak, and it has tails that gradually decrease towards the edges. So, if we look at our histogram here, we see that we have a peak frequency of 12, that's near the middle of our categories, going from 30 to 39 minutes with a midpoint of 34.5. And the frequencies decrease fairly symmetrically on both sides. On the right side, it goes from 10 to 9 to 6, and on the left side, it goes from 12 to 8 to 5. And if we look at the overall shape, it suggests a roughly bell-shaped distribution. So, based on these observations, The histogram suggests a fairly normal distribution. So that's going to be the answer to the second half of our problem. So I hope that this explanation has been useful, and I'll see you in the next video.

In Exercises 9–18, construct the histograms and answer the given questions.
Chicago Commute Time Use the frequency distribution from Exercise 13 in Section 2-1 to construct a histogram. Does it appear to be the graph of data from a population with a normal distribution?

Key Concepts

Histogram

Normal Distribution

Frequency Distribution

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