In Exercises 5–20, find the range, variance, and standard devi...

Question

In Exercises 5–20, find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as “minutes”) in your results. (The same data were used in Section 3-1, where we found measures of center. Here we find measures of variation.) Then answer the given questions.

California Smokers In the California Health Interview Survey, randomly selected adults are interviewed. One of the questions asks how many cigarettes are smoked per day, and results are listed below for 50 randomly selected respondents. How well do the results reflect the smoking behavior of California adults?

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Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to find the range variance and standard deviation of the given sample data. Include appropriate units such as milligrams in your results, then answer the given question. And we're given the situation that says, in a study on caffeine consumption, researchers record the number of milligrams consumed daily by a randomly selected group of adults. The following data represents the daily caffeine intake in milligrams or 5 individuals. How well do these results reflect the caffeine consumption habits of the general population? And we're given the daily caffeine intake in milligrams of the 5 individuals to be 80, 120, 100, 90, and 110. We're also in 4 possible choices as our answers. Your choice A, have range is equal to 40 mg, variance is equal to 125 mg squared, standard deviation is equal to 11.18 mg, and the results are highly representative of the general population since they cover a wide range of values. For choice B, we have the ranges equal to 30 mg, the variance is equal to 250 mg squared. The standard deviation is equal to 15.81 mg, and the results are completely unreliable, and no conclusion can be drawn from such a small data set. For choice C, we have the ranges equal to 20 mg. The variance is equal to 125 mg squared. Standard deviation is equal to 11.18 mg, and the results are perfectly, the results perfectly reflect the caffeine consumption of all adults, as caffeine intake is the same for everyone. And for choice D, we have the range is equal to 40 mg, the variance is equal to 2150 mg squared. The standard deviation is equal to 15.81 mg. And the results provide some insight, but the small sample size. Limits the reliability in representing the general population. Now, the first thing we want to do for this question is to find the range variance and standard deviation of our given data set. So, we'll start off by looking at the range, and recall that the range is just equal to your maximum value of your data set minus the minimum value. So, in this case, if we look at our data set, we see that the maximum value is 120 mg, and we need to subtract from that the minimum value, which is 80 mg. And so, 120 minus 80 is equal to 40, and that has units of milligrams. So that is one of the quantities that we need to find. The next quantity we need to look at is the variance, which is represented as Squad. And we call your formula for the variance, that's going to be equal to the sum. Of the quantity of X by minus X bar in quantity squared divided by the quantity of N minus 1 in quantity. Now, in this case, XI is going to be our individual data point. X bar is going to be the mean of our data set, and N is the number of data points. So, in this case, N is equal to 5, since we have 5 data points, but we'll need to actually calculate our mean value. So, we'll calculate X bar. And recall that for the being, we just need to add up all of our data points and divide by the number of data points that we have. So, in numerator, we'll have the quantity of 80 plus 120, plus 100 plus 90 + 110 in quantity, that gets divided by the number of data points, which is 5. And when we evaluate this expression, this is going to be equal to 100. So now we can actually calculate our variants. So that means that Squad is going to be equal to. In the numerator will have the quantity of 80 minus 100 in quantity squared. Plus, the quantity of 120 minus 100 in quantity squared. Plus the quantity of 100 minus 100 in quantity squared. Plus the quantity of 90 minus 100 in quantity squared, plus. The quantity of 110 minus 100 in quantity squared. And all of that is divided by the quantity of N minus 1. So there'll be the quantity of, of 5 minus 1. So, still find this expression, this is going to be equal to. So, for our first term, we'll have 80 minus 100, which will give us -20, and when we square that, we get 400. Then we'll have plus for the second term in the numerator, we'll have 120 minus 100, which again, is going to give us 20, we square that, it's going to be 400. Then we'll have plus 100 minus 100 is 0, when the square 0, I still get 0. Then we'll have less. I'll have 90 minus 100, which will give me 10, and when we square minus 10, we get 100. And then we'll have plus, for the last term, we will have 1 to 10 minus 100, which will give me 10, and with square 10, I get 100 again. And all of that is divided by quantity of 5 minus 1, which is 4. And evaluating this expression, this is equal to. 250 Milligrams squared. That's because we squared the quantity of X of I minus X bar, each of which have units of milligrams, and so when we square that quantity, we also need to square the units. So this here is the 2nd quantity that we needed to find. Now, for the last quantity, we need to find the standard deviation, and so that is going to be represented by S, and this is just equal to the square root of Squared. Which is equal to the square root of 250. Which is approximately equal to. 15.81, and this has units of milligrams. Since I'll be taking the square root, I also need to take the square root of the units. I want to take the square root of milligrams squared, I get the units of milligrams. So this here is the last quantity that we needed to find. So now we just need to answer our final question. Which is how well do these results reflect the caffeine consumption of a general population. And these results will provide some insight into the variation and caffeine consumption between individuals. However, the small sample size will actually limit the reliability in representing the general population. We just don't have enough data to generalize it to everyone. So based on this information, We look at our answer choices, we see that the correct answer choice corresponds to answer choice D. Since it has the range equal to 40 mg, the variance is equal to 250 mg squared, and the standard deviation equal to 15.81 mg, and also has the correct explanation, which is the results provide some insight, but the small sample size limits the reliability in representing the general population. So, I hope that this explanation has been useful, and I'll see you in the next video.

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Key Concepts

Range

Variance

Standard Deviation

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