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Chapter 3: Problem 16

The following data give the annual salaries (in thousand dollars) of 20 randomly selected health care workers. \(\begin{array}{llllllllll}50 & 71 & 57 & 39 & 45 & 64 & 38 & 53 & 35 & 62 \\ 74 & 40 & 67 & 44 & 77 & 61 & 58 & 55 & 64 & 59\end{array}\) a. Calculate the mean, median, and mode for these data. b. Calculate the \(15 \%\) trimmed mean for these data.

Short Answer

Expert verified
Mean = \$55.9 thousand, Median = \$57.5 thousand, Mode = None, 15% Trimmed Mean = \$56.1 thousand

Step by step solution

01

Compute the Mean

Add up all the annual salaries and divide by the total number, which is 20. That is:\[Mean = \frac{50 + 71 + 57 + 39 + 45 + 64 + 38 + 53 + 35 + 62 +74 + 40 + 67 + 44 + 77 + 61 + 58 + 55 +64 + 59}{20}\]
02

Compute the Median

First, order the data set from low to high. Then, since the total number of data is even (20), the median is the average of the 10th and 11th values. That is \[Median = \frac{10th value + 11th value}{2}\]
03

Compute Mode

Mode is the most frequent value in the data set. In this data set, the mode is not a single value as no value appears more than once.
04

Compute 15% Trimmed Mean

First, remove the lowest 15% and highest 15% of the data. This leaves 70% of the data in the middle. Then, find the mean of this trimmed data by adding up the remaining values and divide by the count. In this case, 15% of 20 entries is 3. So, remove the 3 smallest and 3 largest values and then compute the mean of the remaining 14 values.

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Key Concepts

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

Mean Calculation
To find the mean (or average) of a set of numbers, add up all the numbers and divide by the total quantity of these numbers. This statistic provides a central value that summarizes the data set.
  • First, sum up all the annual salaries: 50 + 71 + 57 + 39 + 45 + 64 + 38 + 53 + 35 + 62 + 74 + 40 + 67 + 44 + 77 + 61 + 58 + 55 + 64 + 59. This results in a total of 1189.
  • Next, count the number of data points, which is 20.
  • Finally, divide the total sum by the number of data points to find the mean: \[Mean = \frac{1189}{20} = 59.45\]
The mean salary for these health care workers is $59,450.
Median Calculation
The median represents the middle value in a data set and provides a measure of central tendency. It requires organizing the data in ascending order first.
  • Order the data: 35, 38, 39, 40, 44, 45, 50, 53, 55, 57, 58, 59, 61, 62, 64, 64, 67, 71, 74, 77.
  • Since there are 20 numbers, and 20 is even, the median is calculated as the average of the 10th and 11th data points.
  • The 10th value is 57, and the 11th value is 58.The median is then calculated as: \[Median = \frac{57 + 58}{2} = 57.5\]
The median salary, therefore, is $57,500, which indicates that about half the workers earn below and half above this amount.
Mode Calculation
The mode is the value that appears most frequently in a data set. It helps identify the most common value, though not all data sets have a mode.
  • Compare each value with the others to determine frequency.
  • In this particular set, each salary occurs only once.
  • Since no salary repeats, this data set has no mode.
Absence of a mode simply means there's no repetitive salary, which implies a diverse range of salaries among the health care workers.
Trimmed Mean
A trimmed mean eliminates a percentage of the smallest and largest values, providing a measure less sensitive to outliers.
  • First, calculate 15% of the 20 salaries: 15% of 20 is 3.
  • Remove the three smallest values (35, 38, 39) and the three largest values (71, 74, 77).
  • This leaves 14 salaries: 40, 44, 45, 50, 53, 55, 57, 58, 59, 61, 62, 64, 64, 67.
  • Add these and divide by 14: \[Trimmed\ Mean = \frac{843}{14} = 60.21\]
Thus, the 15% trimmed mean is $60,210, offering a balanced average that reduces the impact of extreme salaries.

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Most popular questions from this chapter

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, the summary statistics are 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? Using Chebyshev's theorem, what is the lower bound for the percentage of the data that should fall in the interval?

The following data give the number of patients who visited a walk-in clinic on each of 20 randomly selected days. \(\begin{array}{llllllllll}23 & 37 & 26 & 19 & 33 & 22 & 30 & 42 & 24 & 26 \\\ 28 & 32 & 37 & 29 & 38 & 24 & 35 & 20 & 34 & 38\end{array}\) a. Calculate the range, variance, and standard deviation for these data. b. Calculate the coefficient of variation.

Seven airline passengers in economy class on the same flight paid an average of \(\$ 361\) per ticket. Because the tickets were purchased at different times and from different sources, the prices varied. The first five passengers paid \(\$ 420, \$ 210, \$ 333, \$ 695\), and \(\$ 485\). The sixth and seventh tickets were purchased by a couple who paid identical fares. What price did each of them pay?

The following data give the time (in minutes) that each of 20 students selected from a university waited in line at their bookstore to pay for their textbooks in the beginning of the Fall 2015 semester. \(\begin{array}{rrrrrrrrrr}15 & 8 & 23 & 21 & 5 & 17 & 31 & 22 & 34 & 6 \\ 5 & 10 & 14 & 17 & 16 & 25 & 30 & 3 & 31 & 19\end{array}\) Prepare a box-and-whisker plot. Comment on the skewness of these data.

Briefly explain the difference between a population parameter and a sample statistic. Give one example of each.
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