Using the Empirical Rule In Exercises 29–34, ...

Question

Using the Empirical Rule In Exercises 29–34, use the Empirical Rule.

The speeds for eight vehicles are listed. Using the sample statistics from Exercise 29, determine which of the data entries are unusual. Are any of the data entries very unusual? Explain your reasoning.

70, 78, 62, 71, 65, 76, 82, 64

Accepted Answer

All right, hello, everyone. So this question says, using the empirical rule, determine which of the following plant heights and inches are unusual or very unusual. The heights for 8 plants are listed as follows. Assume the sample mean height is 48 inches and the standard deviation is 6 inches. And here we have 4 different entry choices labeled A through D. So first, what does it mean when we refer to data as unusual or very unusual? Well, recall that data is considered to be unusual, if it is higher than 2 standard deviations away from the mean, but Less than or equal to 3 standard deviations. By contrast, a very unusual value is typically. More than 3 standard deviations above the mean. So first, we have to find the range for the unusual and very unusual values. Now, in this case, we already have both the mean and the standard deviation. Me, which is the mean, is equal to 48. And sigma, that's the standard deviation, is equal to 6. So first, Let's find the lower bound. Of the unusual values range. So here In order for X in this case to be considered unusual. X. Should be between Mew subtracted by 3 sigma. And Mew subtracted by 2 sigma. So X Is between 48. Subtracted by 3 multiplied by 6. And 48 Subtracted by 2 multiplied by 6. Here For the lower bound of the unusual value range that's between 30. And 36 inches. And now we can find the upper bound. So now the upper bound Is going to be between. You added 23 sigma. And you added to 2 sigma. So that's 48. Added to 2 multiplied by 6. And 48 added to 3 multiplied by 6. And so for the upper bound, that's between 60. And 66. Inches Now, slight correction here because. X can be equal to 66, but not equal to 60. So recall that this range represents the unusual value. So, for data to be considered very unusual, X should be either less than 30 inches or greater than 66 inches. And now we can identify those heights that are either unusual or very unusual. And here let me scroll up to see that data set. Now, looking at the values themselves, 61 in particular is noteworthy, because notice that 61 is between 60 and 66. That's the upper bound of the unusual range. So that qualifies as an unusual value. As for very unusual heights, all of the values given are between 30 and 66 inches. Which means that none of them can be considered very unusual. So Our correct answer is ultimately going to be option C in the multiple choice, because only the number 61 is considered unusual, and none were considered very unusual. And there you have it. So with that being said, thank you so very much for watching, and I hope you found this helpful.

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Key Concepts

Empirical Rule

Standard Deviation

Unusual Data Points

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