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Chapter 4: Problem 24

Give two sets of five numbers that have the same mean but different standard deviations, and give two sets of five numbers that have the same standard deviation but different means.

Short Answer

Expert verified
Two sets of numbers with the same mean but different standard deviations could be [1, 2, 3, 4, 5] and [2, 2, 3, 4, 5]. Two sets of numbers with the same standard deviation but different means could be [2, 4, 6, 8, 10] and [3, 5, 7, 9, 11].

Step by step solution

01

Find two sets of numbers with the same mean but different standard deviations

Let's generate two sets of numbers with the same mean. For example, set A = [1, 2, 3, 4, 5] and set B = [2, 2, 3, 4, 5]. Both sets have a mean of 3 ((1+2+3+4+5)/5 = 3 for set A and (2+2+3+4+5)/5 = 3 for set B). However, they have different standard deviations. The standard deviation of set A will be higher than that of set B as it contains more spread out numbers.
02

Calculate the standard deviations

Calculate the standard deviation for both sets. The standard deviation for set A is approximately 1.41, while for set B, it is lower at approximately 1.10, confirming that they are indeed different.
03

Find two sets of numbers with the same standard deviation but different means

Now let's find two sets of numbers with the same standard deviation but different means. For example, set C = [2, 4, 6, 8, 10] and set D = [3, 5, 7, 9, 11]. Both sets have a standard deviation of approximately 2.83, but different means. The mean of set C is 6, while the mean of set D is 7.
04

Calculate the means

Calculate the mean for both sets. The mean for set C=(2+4+6+8+10)/5 = 6 and for set D it is (3+5+7+9+11)/5 = 7, showing that the means are different.

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Most popular questions from this chapter

The amount of aluminum contamination (in parts per million) in plastic was determined for a sample of 26 plastic specimens, resulting in the following data ("The Log Normal Distribution for Modeling Quality Data When the Mean is Near Zero." Journal of Quality Technology \([1990]: 105-110)\) : \(\begin{array}{rrrrrrrrr}30 & 30 & 60 & 63 & 70 & 79 & 87 & 90 & 101 \\ 102 & 115 & 118 & 119 & 119 & 120 & 125 & 140 & 145 \\ 172 & 182 & 183 & 191 & 222 & 244 & 291 & 511 & \end{array}\) Construct a boxplot that shows outliers, and comment on the interesting features of this plot.

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