In Exercises 17 and 18, (a) construct a binomial distribution,...

Question

In Exercises 17 and 18, (a) construct a binomial distribution, (b) graph the binomial distribution using a histogram and describe its shape, and (c) identify any values of the random variable x that you would consider unusual. Explain your reasoning.

Seventy-two percent of U.S adults have read a book in any format in the past year. You randomly select five U.S adults and ask them whether they have read a book in any format in the past year. The random variable represents the number of adults who have read a book in any format in the past year. (Source: Pew Research)

Accepted Answer

All right, hello, everyone. So this question says a recent survey found that 72% of workers bring their own lunch to work at least 3 days a week. You randomly choose 5 workers and ask each one if they bring their own lunch at least 3 days a week. Let the random variable X represent the number of workers who do this. Create a binomial distribution and a corresponding histogram to answer the following. For part one, what is the histogram shape? For part two, which value of X would be considered unusual? And for part 3, why is this value considered unusual? So, based on the information that we are given, right, we can see that, or excuse me, lowercase n, which is the number of trials, is equal to 5. And P, which is the probability of success, is equal to 0.72. Which means that Q or the probability of failure. is equal to 1 subtracted by P, which equals 0.28. Now, with this information, and knowing that this is a binomial distribution, we can use the binomial probability formula. Which states that P of X. is equal to And Z Multiplied by P to the power of X. And multiplied By Q. To the power of n subtracted by X. So, with this information in mind, I went ahead and made a table with the possible values of X ranging from 0 to 5 on the left, and space on the right for the corresponding probability value. So, P X is obtained by plugging in each value of X into the probability formula for the binomial distribution. And when you do so, the values are as follows. 4 X equals 0, P is 0.0017. For 1, that's 0.0221. For 2, that's 0.1138. For 3, that's 0.2926. For 4, that's 0.3762. And for 5, that's 0.1935. So now this is the data that you would plug in. To create the histogram. So the X axis would showcase the number of workers, and the probability values for each would be shown on the Y axis. When you do so, the resulting histogram should look like this, and let me bring it up to the screen here. So now that you're looking at the histogram itself, you can see that most of the data points are clustered towards the right side of the histogram. And because the data is clustered towards the right, the tail of the histogram. Is longer on the left. So because the tail of the histogram is longer on the left. This makes the histogram left skewed. Which answers part one of the question. So now, what are the unusual values and why are they unusual? Well, recall that a value is considered to be unusual if the probability. Or that value is less than 0.05. So going back to the table that we constructed with the values of PF X. The X values with probabilities less than 0.05 are 0. And one Which means that these are unusual values. So now we have the answers to parts 12, and 3 of the question. And there you have it. So with that being said, thank you so very much for watching, and I hope you found this helpful.

Key Concepts

Binomial Distribution

Histogram

Unusual Values

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