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Chapter 4: Problem 16

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
The two sets of numbers with the same mean but different standard deviations are [1,2,3,4,5] and [3,3,3,3,3]. The two sets of numbers that have the same standard deviation but different means are [1,2,3,4,5] and [6,7,8,9,10].

Step by step solution

01

Finding two sets with same mean but different standard deviations

To have the same average, the sum of the numbers in each set should be equal. Let's make two sets: [1,2,3,4,5] and [3, 3, 3, 3, 3]. Both have the mean (average) of 3 but the second set doesn't vary from the mean, hence it has a standard deviation of 0 while the first set varies from the mean and therefore has a greater standard deviation.
02

Finding two sets with same standard deviation but different means

To have the same standard deviation, the numbers in each set should differ from mean similarly. Let's consider two sets: [1,2,3,4,5] and [6,7,8,9,10]. Both sets have the same spread or variation from the mean, hence the same standard deviation. However, they have different central locations and therefore different mean values.

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Most popular questions from this chapter

The article "Taxable Wealth and Alcoholic Beverage Consumption in the United States" (Psychological Reports [1994]: \(813-814\) ) reported that the mean annual adult consumption of wine was \(3.15\) gal and that the standard deviation was \(6.09\) gal. Would you use the Empirical Rule to approximate the proportion of adults who consume more than \(9.24\) gal (i.e., the proportion of adults whose consumption value exceeds the mean by more than 1 standard deviation)? Explain your reasoning.

The accompanying data on milk volume (in grams per day) were taken from the paper "Smoking During Pregnancy and Lactation and Its Effects on Breast Milk Volume" (American Journal of Clinical Nutrition [1991]: \(1011-1016)\) : \(\begin{array}{lllrrll}\text { Smoking } & 621 & 793 & 593 & 545 & 753 & 655 \\\ \text { mothers } & 895 & 767 & 714 & 598 & 693 & \\ \text { Nonsmoking } & 947 & 945 & 1086 & 1202 & 973 & 981 \\ \text { mothers } & 930 & 745 & 903 & 899 & 961 & \end{array}\) Compare and contrast the two samples.

Houses in California are expensive, especially on the Central Coast where the air is clear, the ocean is blue, and the scenery is stunning. The median home price in San Luis Obispo County reached a new high in July 2004 soaring to \(\$ 452,272\) from \(\$ 387,120\) in March 2004. (San Luis Obispo Tribune, April 28,2004\()\). The article included two quotes from people attempting to explain why the median price had increased. Richard Watkins, chairman of the Central Coast Regional Multiple Listing Services was quoted as saying "There have been some fairly expensive houses selling, which pulls the median up." Robert Kleinhenz, deputy chief economist for the California Association of Realtors explained the volatility of house prices by stating: "Fewer sales means a relatively small number of very high or very low home prices can more easily skew medians." Are either of these statements correct? For each statement that is incorrect, explain why it is incorrect and propose a new wording that would correct any errors in the statement.

1 The San Luis Obispo Telegram-Tribune (October 1. 1994) reported the following monthly salaries for supervisors from six different counties: \(\$ 5354\) (Kern), \(\$ 5166\) (Monterey), \$4443 (Santa Cruz), \$4129 (Santa Barbara), \$2500 (Placer), and \$2220 (Merced). San Luis Obispo County supervisors are supposed to be paid the average of the two counties among these six in the middle of the salary range. Which measure of center determines this salary, and what is its value? Why is the other measure of center featured in this section not as favorable to these supervisors (although it might appeal to taxpayers)?

A sample of \(n=5\) college students yielded the following observations on number of traffic citations for a moving violation during the previous year: $$ x_{1}=1 \quad x_{2}=0 \quad x_{3}=0 \quad x_{4}=3 \quad x_{5}=2 $$ Calculate \(s^{2}\) and \(s\). 4.16 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.
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