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Chapter 3: Problem 14

The following data give the 2008 profits (in millions of dollars) of the six Arizona-based companies for the year 2008 (Fortune, May 5,2008 ). The data represent the following companies, respectively: Freeport-McMoRan Copper \& Gold, Avnet, US Airways Group, Allied Waste Industries, Insight Enterprises, and PetSmart. \(2977.0\) \(\begin{array}{lllll}393.1 & 427.0 & 273.6 & 77.8 & 258.7\end{array}\) Find the mean and median for these data. Do these data have a mode?

Short Answer

Expert verified
The mean of these data is approximately 899.55 million dollars, the median is approximately 410.05 million dollars and these data have no mode as no profit value repeats.

Step by step solution

01

Calculate the Mean

To calculate the mean of these data, sum up all the values and then divide by the count of the values. In this case there are six values corresponding to six companies: \(2977.0, 393.1, 427.0, 273.6, 77.8, 258.7\). The mean is found by using the formula: \[ Mean = \frac{sum of values }{count of values}\]
02

Calculate the Median

The median is the middle number in a sorted list. If the count of numbers is even, the median is the average of the two middle numbers. These data are already in descending order. Now, find the middle value. Here, with six numbers, it is the average of the third and fourth values, which are 427.0 and 393.1 respectively.
03

Determine the Mode

The mode is the most frequently occurring value in a dataset. Observing the given data, there's no value that repeats; hence, there is no mode in this dataset.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean Calculation
The mean, often referred to as the average, provides a central value for a data set by summarily expressing all the data points as one single value. To calculate the mean, follow these simple steps:
  • Add up all the numbers in the data set.
  • Count how many numbers are there in the data set.
  • Divide the sum by the total count of numbers.
In this case, for the profits of the companies: - Add the profits: \( 2977.0 + 393.1 + 427.0 + 273.6 + 77.8 + 258.7 \)- The sum is \(4407.2 \).- Then, divide the total by the number of companies, which is six: \[Mean = \frac{4407.2}{6} = 734.53\] So, the mean profit of these six companies is approximately \(734.53\) million dollars.
Median Calculation
The median gives you an understanding of the middle point of a data set. It is less affected by extremely large or small values, making it a reliable measure of central tendency. For finding the median with an even number of data points:
  • First, arrange the data in either ascending or descending order, if not already done.
  • If there is an odd number of data points, select the middle one.
  • If the number is even, find the average of the two central numbers.
In our exercise, the data is already sorted in descending order. For six data points, identify the two numbers in the middle: 427.0 and 393.1.
Calculate the average of these two numbers to get the median:\[Median = \frac{427.0 + 393.1}{2} = 410.05\] Therefore, the median profit is \(410.05\) million dollars. This value represents the mid-range in size of the company profits.
Mode Determination
The mode is the value that appears most frequently in a data set. It can be particularly useful to know if you're examining categorical data or when values repeat themselves. Steps for determining the mode:
  • List each value in the data set.
  • Count how many times each value appears.
  • Identify the value that appears the most.
For the given company's profit data, observe the list as follows: - 2977.0 - 393.1 - 427.0 - 273.6 - 77.8 - 258.7
The values in the dataset are all unique, and no number repeats itself. Hence, this dataset is described as "modaless," meaning it does not have a mode. Having no mode is a common situation when all values are distinct or have the same frequency, which in this case is once.

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Most popular questions from this chapter

The mean life of a certain brand of auto batteries is 44 months with a standard deviation of 3 months. Assume that the lives of all auto batteries of this brand have a bell-shaped distribution. Using the empirical rule, find the percentage of auto batteries of this brand that have a life of a. 41 to 47 months b. 38 to 50 months c. 35 to 53 months

The mean time taken by all participants to run a road race was found to be 220 minutes with a standard deviation of 20 minutes. Using Chebyshev's theorem, find the percentage of runners who ran this road race in a. 180 to 260 minutes b. 160 to 280 minutes c. 170 to 270 minutes

Although the standard workweek is 40 hours a week, many people work a lot more than 40 hours a week. The following data give the numbers of hours worked last week by 50 people. \(\begin{array}{llllllllll}40.5 & 41.3 & 41.4 & 41.5 & 42.0 & 42.2 & 42.4 & 42.4 & 42.6 & 43.3 \\ 43.7 & 43.9 & 45.0 & 45.0 & 45.2 & 45.8 & 45.9 & 46.2 & 47.2 & 47.5 \\ 47.8 & 48.2 & 48.3 & 48.8 & 49.0 & 49.2 & 49.9 & 50.1 & 50.6 & 50.6 \\ 50.8 & 51.5 & 51.5 & 52.3 & 52.3 & 52.6 & 52.7 & 52.7 & 53.4 & 53.9 \\\ 54.4 & 54.8 & 55.0 & 55.4 & 55.4 & 55.4 & 56.2 & 56.3 & 57.8 & 58.7\end{array}\) a. The sample mean and sample standard deviation for this data set are \(49.012\) and \(5.080\), respectively. Using Chebyshev's theorem, calculate the intervals that contain at least \(75 \%, 88.89 \%\), and \(93.75 \%\) of the data. b. Determine the actual percentages of the given data values that fall in each of the intervals that you calculated in part a. Also calculate the percentage of the data values that fall within one standard deviation of the mean. c. Do you think the lower endpoints provided by Chebyshev's theorem in part a are useful for this problem? Explain your answer. d. Suppose that the individual with the first number (54.4) in the fifth row of the data is a workaholic who actually worked \(84.4\) hours last week and not \(54.4\) hours. With this change now \(\bar{x}=49.61\) and \(s=7.10\). Recalculate the intervals for part a and the actual percentages for part b. Did your percentages change a lot or a little? e. How many standard deviations above the mean would you have to go to capture all 50 data values? What is the lower bound for the percentage of the data that should fall in the interval, according to the Chebyshev theorem.

Consider the following two data sets. \(\begin{array}{llllrl}\text { Data Set I: } & 12 & 25 & 37 & 8 & 41 \\ \text { Data Set II: } & 19 & 32 & 44 & 15 & 48\end{array}\) 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.

Briefly explain Chebyshev's theorem and its applications.
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