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Chapter 3: Problem 1

Explain how the value of the median is determined for a data set that contains an odd number of observations and for a data set that contains an even number of observations.

Short Answer

Expert verified
For an odd number data set, the median is the middlemost number when listed in ascending or descending order. For an even number data set, the median is the arithmetic mean of the two middlemost numbers in the ordered set.

Step by step solution

01

Understanding the Median for odd amount of observations

To determine the median for a data set with an odd amount of observations, this set needs to be organized in ascending or descending order. The median is then the value that is exactly in the middle of this ordered data set. This middle value will separate the data set in two halves, having an equal amount of values on each side.
02

Calculating the Median for odd amount of observations

Once the data set is ordered, find out the total number \( n \) of the observations. The position (index) of the median will be \( (n+1) / 2 \). Hence the median is the number at this position in the ordered data set.
03

Understanding the Median for even amount of observations

For a data set with an even amount of observations, there won't be a single middle number. Instead, there will be two middle numbers. According to this, once the data set is ordered, the two middlemost numbers need to be identified.
04

Calculating the Median for even amount of observations

After identifying the two middle numbers at positions \( n / 2 \) and \( (n/2) + 1 \) in the ordered set, the median is calculated as the arithmetic mean of these two numbers, which means it is their sum divided by 2.

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Most popular questions from this chapter

Twenty randomly selected married couples were asked how long they have been married. Their responses (rounded to years) are listed below. \(\begin{array}{llrllrrlll}12 & 27 & 8 & 15 & 5 & 9 & 18 & 13 & 35 & 23 \\ 19 & 33 & 41 & 59 & 3 & 26 & 5 & 34 & 27 & 51\end{array}\) a. Calculate the mean, median, and mode for these data. b. Calculate the \(10 \%\) trimmed mean for these data.

Consider the following two data sets. Data Set I: \(\begin{array}{lllrl}12 & 25 & 37 & 8 & 41 \\ 19 & 32 & 44 & 15 & 48\end{array}\) Data Sel \(1 .\) Data Set II: Note that each value of the second data set is obtained by adding 7 to the corresponding value of the first data set. Calculate the standard deviation for each of these two data sets using the formula for sample data. Comment on the relationship between the two standard deviations.

A sample of 2000 observations has a mean of 74 and a standard deviation of \(12 .\) Using Chebyshev's theorem, find the minimum percentage of the observations that fall in the intervals \(\bar{x} \pm 2 s, \bar{x} \pm 2.5 s\), and \(\bar{x} \pm 3 s\). Note that \(\bar{x} \pm 2 s\) represents the interval \(\bar{x}-2 s\) to \(\bar{x}+2 s\), and so on.

A survey of young people's shopping habits in a small city during the summer months of 2015 showed the following: Shoppers aged 12 to 14 years took an average of 8 shopping trips per month and spent an average of \(\$ 14\) per trip. Shoppers aged 15 to 17 years took an average of 11 trips per month and spent an average of \(\$ 18\) per trip. Assume that this city has 1100 shoppers aged 12 to 14 years and 900 shoppers aged 15 to 17 years. a. Find the total amount spent per month by all these 2000 shoppers in both age groups. b. Find the mean number of shopping trips per person per month for these 2000 shoppers. c. Find the mean amount spent per person per month by shoppers aged 12 to 17 years in this city.

Actuaries at an insurance company must determine a premium for a new type of insurance. A random sample of 40 potential purchasers of this type of insurance were found to have incurred the following losses (in dollars) during the past year. These losses would have been covered by the insurance if it were available. \(\begin{array}{rrrrrrrrrr}100 & 32 & 0 & 0 & 470 & 50 & 0 & 14,589 & 212 & 93 \\ 0 & 0 & 1127 & 421 & 0 & 87 & 135 & 420 & 0 & 250 \\ 12 & 0 & 309 & 0 & 177 & 295 & 501 & 0 & 143 & 0 \\ 167 & 398 & 54 & 0 & 141 & 0 & 3709 & 122 & 0 & 0\end{array}\) a. Find the mean, median, and mode of these 40 losses. b. Which of the mean, median, or mode is largest? c. Draw a box-and-whisker plot for these data, and describe the skewness, if any. d. Which measure of center should the actuaries use to determine the premium for this insurance?
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