Question

determine whether each statement makes sense or does not make sense, and explain your reasoning. im working with data sets with different means and the same standard deviation. choose the correct answer below. a. the statement makes sense. data sets can have different means and the same standard deviation. for example, if two data sets contained the values {1,3} and {2,4}, the means for each would be 2 and 3 respectively, but they both have the same standard deviation of approximately 1.41421. b. the statement does not make sense. data sets can never have the same standard deviation unless they are the same data set, in which case their means would be the same. c. the statement does not make sense. if data sets have different means, then they must have different standard deviations. d. the statement makes sense. the mean of a data set is completely unrelated to the standard deviation of a data set. thus data sets can have different means and the same standard deviation.

Explanation:

Let's analyze each option:

• Option A:
• Calculate the mean and standard deviation for the data set \(\{1,3\}\).
• Mean: \(\bar{x}_1=\frac{1 + 3}{2}=2\)
• Variance: \(s_1^2=\frac{(1 - 2)^2+(3 - 2)^2}{2-1}=\frac{(-1)^2+(1)^2}{1}=2\), standard deviation \(s_1=\sqrt{2}\approx1.4142\)
• Calculate the mean and standard deviation for the data set \(\{2,4\}\).
• Mean: \(\bar{x}_2=\frac{2+4}{2}=3\)
• Variance: \(s_2^2=\frac{(2 - 3)^2+(4 - 3)^2}{2-1}=\frac{(-1)^2+(1)^2}{1}=2\), standard deviation \(s_2=\sqrt{2}\approx1.4142\)
• This shows that data sets can have different means and the same standard deviation.
• Option B:
• As shown in the example above, different data sets can have the same standard deviation. So, the claim that data sets can never have the same standard deviation unless they are the same data set is wrong.
• Option C:
• The example in Option A refutes this. Different means do not imply different standard deviations.
• Option D:
• While the mean and standard deviation are not directly determined by each other (they measure different aspects - central tendency and spread respectively), Option A provides a more concrete and correct reasoning with an example.

Answer:

A. The statement makes sense. Data sets can have different means and the same standard deviation. For example, if two data sets contained the values \(\{1,3\}\) and \(\{2,4\}\), the means for each would be \(2\) and \(3\) respectively, but they both have the same standard deviation of approximately \(1.4142\)