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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

In a study investigating the effect of car speed on accident severity, 5000 reports of fatal automobile accidents were examined, and the vehicle speed at impact was recorded for each one. For these 5000 accidents, the average speed was \(42 \mathrm{mph}\) and the standard deviation was 15 mph. A histogram revealed that the vehicle speed at impact distribution was approximately normal. a. Roughly what proportion of vehicle speeds were between 27 and \(57 \mathrm{mph} ?\) b. Roughly what proportion of vehicle speeds exceeded \(57 \mathrm{mph} ?\)

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The Los Angeles Times (July 17. 1995) reported that in a sample of 364 lawsuits in which punitive damages were awarded, the sample median damage award was \(\$ 50,000\), and the sample mean was \(\$ 775,000\). What does this suggest about the distribution of values in the sample?

The following data values are 2009 per capita expenditures on public libraries for each of the 50 U.S. states (from www.statemaster.com): \(\begin{array}{rrrrrrr}16.84 & 16.17 & 11.74 & 11.11 & 8.65 & 7.69 & 7.48 \\\ 7.03 & 6.20 & 6.20 & 5.95 & 5.72 & 5.61 & 5.47 \\ 5.43 & 5.33 & 4.84 & 4.63 & 4.59 & 4.58 & 3.92 \\ 3.81 & 3.75 & 3.74 & 3.67 & 3.40 & 3.35 & 3.29 \\ 3.18 & 3.16 & 2.91 & 2.78 & 2.61 & 2.58 & 2.45 \\ 2.30 & 2.19 & 2.06 & 1.78 & 1.54 & 1.31 & 1.26 \\ 1.20 & 1.19 & 1.09 & 0.70 & 0.66 & 0.54 & 0.49\end{array}\) a. Summarize this data set with a frequency distribution. Construct the corresponding histogram. b. Use the histogram in Part (a) to find approximate values of the following percentiles: i. 50 th iv. 90 th ii. \(\quad 70\) th v. 40 th iii. 10 th
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