Data Set A: The horizontal axis is labeled Integer. It ranges from 10 to 60 and is divided into 5

Question

Two data sets of $23$ integers each are summarized in the histograms shown. For each of the histograms, the first interval represents the frequency of integers greater than or equal to $10$ , but less than $20$ . The second interval represents the frequency of integers greater than or equal to $20$ , but less than $30$ , and so on. What is the smallest possible difference between the mean of data set A and the mean of data set B?

Accepted Answer

Choice B is correct. The histograms shown have the same shape, but data set A contains values between 20 and 60 and data set B contains values between 10 and 50 . Thus, the mean of data set A is greater than the mean of data set B. Therefore, the smallest possible difference between the mean of data set A and the mean of data set B is the difference between the smallest possible mean of data set A and the greatest possible mean of data set B. In data set A, since there are 3 integers in the interval greater than or equal to 20 but less than 30 , 4 integers greater than or equal to 30 but less than 40 , 7 integers greater than or equal to 40 but less than 50 , and 9 integers greater than or equal to 50 but less than 60 , the smallest possible mean for data set A is 3 /mo> 20 + 4 /mo> 30 + 7 /mo> 40 + 9 /mo> 50 23 . In data set B, since there are 3 integers greater than or equal to 10 but less than 20 , 4 integers greater than or equal to 20 but less than 30 , 7 integers greater than or equal to 30 but less than 40 , and 9 integers greater than or equal to 40 but less than 50 , the largest possible mean for data set B is 3 /mo> 19 + 4 /mo> 29 + 7 /mo> 39 + 9 /mo> 49 23 . Therefore, the smallest possible difference between the mean of data set A and the mean of data set B is 3 /mo> 20 + 4 /mo> 30 + 7 /mo> 40 + 9 /mo> 50 23 - 3 /mo> 19 + 4 /mo> 29 + 7 /mo> 39 + 9 /mo> 49 23 , which is equivalent to 3 /mo> 20 - 3 /mo> 19 + 4 /mo> 30 - 4 /mo> 29 + 7 /mo> 40 - 7 /mo> 39 + 9 /mo> 50 - 9 /mo> 49 23 . This expression can be rewritten as 3 20 - 19 + 4 30 - 29 + 7 40 - 39 + 9 50 - 49 23 , or 23 23 , which is equal to 1 . Therefore, the smallest possible difference between the mean of data set A and the mean of data set B is 1 . Choice A is incorrect. This is the smallest possible difference between the ranges, not the means, of the data sets. Choice C is incorrect. This is the difference between the greatest possible mean, not the smallest possible mean, of data set A and the greatest possible mean of data set B. Choice D is incorrect. This is the smallest possible difference between the sum of the values in data set A and the sum of the values in data set B, not the smallest possible difference between the means.

Question a4fc06