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

Question

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

Hershey’s Kisses Use the frequency distribution from Exercise 20 in Section 2-1 to construct a histogram. In using a strict interpretation of the criteria for being a normal distribution, does the histogram appear to depict data from a population with a normal distribution?

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to create a histogram using the frequency distribution of coffee brewing times in seconds shown below. Does the histogram appear to show data from a normal distribution when the requirements for normal distribution are strictly applied? And below the text, we're given a table that has two columns. In the first column, it has brewing times in seconds, and it has the intervals of 40 to 49, 50 to 59, 60 to 69, 70 to 79, and 80 to 89. And in the second column, it's labeled as frequencies, and it has the values of 36, 12, 9, and 4. Now, the first thing we need to do is create a histogram, and what we're going to do is actually calculate the midpoint for each of our booing time intervals. So, we call that to calculate the midpoint of an interval. That is going to be equal to the upper bound, which will label as UB plus the lower bound, which we will label as LB, and that quantity is going to be divided by 2. So, in our table here, we're going to create a 3rd column. That is labeled as midpoint. And we're gonna calculate the midpoint of each of our time intervals. So, for the time interval from 40 to 49 seconds, when we calculate its midpoint, we get 44.5. For 50 to 59, we have a midpoint of 54.5. 4602 69, the midpoint is going to be 64.5. 472 79, the midpoint is 74.5. And for the interval of 80 to 89 seconds, the midpoint is going to be 84.5. So now what we're going to do is use our midpoints and frequencies to create a histogram. So to create a histogram, we can either use a spreadsheet software or draw it manually on graph paper. But in either case, on the x-axis, we'll plot our class midpoints, and on the Y axis, we'll plot the corresponding frequencies. And to do that, we'll draw a vertical bar with heights equal to the frequencies of each class. So doing so, we get the following graph. Where on the X-axis, we have plotted our midpoint categories of 44.5, 54.5, 64.5, 74.5, and 84.5. And on the y axis we have the frequencies, and we have drawn a vertical bar corresponding to the frequency for each of our midpoint classes. And so for 44.5, we have a frequency of 3. For 54.5, we have a frequency of 6.4 64.5, we have a frequency of 12, for 74.5, we have a frequency of 9, and for 84.5, we have a frequency of 4. So this here is the histogram that the question was looking for. Now, to answer the second half of this question, we need to determine whether a histogram represents data from a normal distribution. And so we call that for a normal distribution that the histogram should have a single peak. The normal distribution should be symmetric about its center, and for a normal distribution, it should not be skewed. So that means it should have roughly equal tails on both sides. And if we look at our histogram, we see that it's somewhat bell shaped, but is slightly skewed to the right. And since it is not perfectly symmetrical, it does not strictly meet the criteria for normality. Therefore, the answer to the second half of this question is going to be no. So I hope that this explanation has been useful, and I'll see you in the next video.

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Key Concepts

Histogram

Normal Distribution

Criteria for Normality

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