Constructing and Graphing Binomial Distributions In Exercises ...

Question

Constructing and Graphing Binomial Distributions In Exercises 27–30, (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.

Workplace Cleanliness Fifty-seven percent of employees judge their peers by the cleanliness of their workspaces. You randomly select 10 employees and ask them whether they judge their peers by the cleanliness of their workspaces. The random variable represents the number who judge their peers by the cleanliness of their workspaces. (Source: Adecco)

Accepted Answer

All right. Hello, everyone. So this question says, 62% of employees prefer email as their primary method of workplace communication. You randomly select 10 employees and ask whether they prefer email as their primary communication method. The random variable X represented the number of employees who prefer email. Question one says what is the most appropriate description of the histogram's shape. For part two, which value of X is considered unusual or would be considered unusual? And for part 3, why is this value considered unusual? So, first and foremost, from the information that we're given, we can see that P, which is the probability of success in this context, is equal to 0.62. Now, the probability of failure, that is the probability that the employee does not prefer email, would be one subtracted by P. So one subtracted by 0.62 would give you 0.38. And in this case, we also have the number of trials or lower case n, which is equal to 10 for the number of employees that were selected. So Recall that the binomial probability formula is P X. Is equal to NZ. Multiplied by P to the power of X. And multiplied by 1 subtracted by P to the power of N subtracted by Z. So here On the left side of the screen, I went ahead and pre-made a table with each possible value of X ranging from 0 to 10, and then space on the right column for the corresponding probability value, which you would obtain by plugging in each value of X to. The binomial probability formula. And so when you do so, the values are as follows. For 0, that's 0.0001. For one, that's 0.0010. For 2, that's 0.0075. For 3, that's 0.0327. For 4, that's 0.0934. For 5, that's 0.1829. For 6, that's 0.2487. 47, that's 0.2319. 48. That's 0.1419. For 9, that's 0.0514. And for 10, that's 0.0084. And so now that you have the corresponding probabilities for each value of X, you would go ahead and plot your histogram. So, in this case, the number of employees who prefer email would be the X axis, and their probabilities would be shown on the Y axis. When you do this, the histogram should look like what I'm about to show you on the screen here. And let me actually make this bigger. Just for visibility. So, when you consider the shape of the resulting histogram, you can see that most of the data. I skewed towards the right side of the histogram itself. So, even though the majority of the data is clustered towards the right side, clustered towards the higher values, you can see that the tail of the histogram. Is longer on the left side. And the reason why the tail is longer on the left is because that data is skewed towards. Or are clustered rather towards the right. So because of that, the data has a left SKU. Because the tail is longer on the left side. So, now that we've discussed the shape of the histogram, what are the unusual values of X, and why are they unusual? Well, recall that an outcome is considered to be unusual if the probability is less than 0.05. So going back to the values for P X that we've calculated earlier, the unusual ones. That is, those with probabilities less than 0.05 would be 0. One 2 3 And 10 And once again, the reason why they're unusual is because their probabilities are less than 0.05. So now all three parts of the question have been solved. And 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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