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Chapter 3: Problem 118

Assume that the annual earnings of all employees with CPA certification and 6 years of experience and working for large firms have a bell-shaped distribution with a mean of \(\$ 134,000\) and a standard deviation of \(\$ 12,000\). a. Using the empirical rule, find the percentage of all such employees whose annual earnings are hetween i. \(\$ 98,000\) and \(\$ 170,000\) ii. \(\$ 110,000\) and \(\$ 158,000\) "b. Using the empirical rule, find the interval that contains the annual earnings of \(68 \%\) of all such employees.

Short Answer

Expert verified
a.i. Using the empirical rule, about 99.7% of employees earn between \$98,000 and \$170,000 per year.\na.ii. Using the empirical rule, about 95% of employees earn between \$110,000 and \$158,000 per year.\nb. The range of annual earnings for 68% of these employees is between \$122,000 and \$146,000.

Step by step solution

01

Understanding Empirical Rule

The Empirical Rule states that in a bell-shaped symmetrical distribution: \n\n- 68% of the data falls within one standard deviation of the mean.\n- 95% falls within two standard deviations.\n- Almost all (99.7%) falls within three standard deviations.\n Here the mean is \$134,000 and the standard deviation is \$12,000.
02

Calculation for Range \$98,000 and \$170,000

This range includes values that are from three standard deviations below the mean (\$134,000 - 3*\$12,000 = \$98,000) to three standard deviations above the mean (\$134,000 + 3*\$12,000 = \$170,000). So according to the empirical rule, about 99.7% of all data falls in this range.
03

Calculation for Range \$110,000 and \$158,000

This range includes values that are from two standard deviations below the mean (\$134,000 - 2*\$12,000 = \$110,000) to two standard deviations above the mean (\$134,000 + 2*\$12,000 = \$158,000). So according to the empirical rule, about 95% of all data falls in this range.
04

Determine the Earnings Interval for 68% of Employees

Using the empirical rule, 68% of employee earnings will fall within one standard deviation of the mean. One standard deviation below the mean is \$134,000 - \$12,000 = \$122,000 and one standard deviation above the mean is \$134,000 + \$12,000 = \$146,000. Therefore, 68% of employees earn between \$122,000 and \$146,000 per year.

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Key Concepts

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

Bell-Shaped Distribution
A bell-shaped distribution, also known as a normal distribution, is one of the most common patterns observed in statistics. It is symmetric, meaning it looks the same on both sides of the center.
This shape often occurs naturally in various fields, such as finance or biology. The center peak represents the average, and data tails off symmetrically on both sides. This pattern makes understanding and predicting data behaviors much easier.
The bell curve implies that most data points are around the average, with fewer appearing as you move away from the center.
  • This means extremes (very high or very low values) are rare.
  • The distribution is used extensively because of its mathematical properties and usefulness in real-world scenarios.
In the case of CPA-certified employees' earnings, the bell shape helps us apply specific rules like the Empirical Rule to make accurate predictions about data spread.
Mean and Standard Deviation
The mean and standard deviation are crucial statistical measures that describe a dataset.
The mean, often referred to as the average, is calculated by adding all data points and dividing by the number of points. It gives us a central value in the dataset.
For our CPA earnings example, the mean is $134,000, suggesting this is the typical earning for an employee with CPA certification and 6 years of experience.
  • The mean provides a quick overview of the entire dataset's "center".
The standard deviation measures the data's spread around the mean. The larger the standard deviation, the more spread out the data. For CPA employees, the standard deviation is $12,000.
  • This indicates that most employee earnings are within $12,000 of the mean.
Understanding these two metrics helps in assessing the variability and center of the data, making them critical for applying the Empirical Rule.
Percentage of Data Within Standard Deviations
The Empirical Rule is a statistical guideline that describes how data in a bell-shaped distribution spreads around the mean. It tells us the percentage of data within specific standard deviation boundaries.
  • 68% of data falls within one standard deviation of the mean. For example, in our case: From $122,000 to $146,000.
  • 95% of data fits within two standard deviations. Here, it's between $110,000 to $158,000.
  • 99.7% of data fits within three standard deviations. For employee earnings, that's between $98,000 and $170,000.
The rule gives us a quick way to predict where most of the data will lie without calculating each piece.
This method helps in determining normal behavior versus any significant outliers in the data. For educators and students, understanding this distribution spread is crucial for analyzing various statistical data effectively.

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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 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.

The following data give the numbers of new cars sold at a dealership during a 20 -day period. \(\begin{array}{lrlrlllll}8 & 5 & 12 & 3 & 9 & 10 & 6 & 12 & 8 \\\ 4 & 16 & 10 & 11 & 7 & 7 & 3 & 5 & 9\end{array}\) a. Calculate the values of the three quartiles and the interquartile range. Where does the value of 4 lie in relation to these quartiles? b. Find the (approximate) value of the 25 th percentile. Give a brief interpretation of this percentile. c. Find the percentile rank of 10 . Give a brief interpretation of this percentile rank.

The heights of five starting players on a basketball team have a mean of 76 inches, a median of 78 inches, and a range of 11 inches. a. If the tallest of these five players is replaced by a substitute who is 2 inches taller, find the new mean, median, and range. b. If the tallest player is replaced by a substitute who is 4 inches shorter, which of the new values (mean, median, range) could you determine, and what would their new values be?

Suppose that there are 150 freshmen engineering majors at a college and each of them will take the same five courses next semester. Four of these courses will be taught in small sections of 25 students each, whereas the fifth course will be taught in one section containing all 150 freshmen. To accommodate all 150 students, there must be six sections of each of the four courses taught in 25 -student sections. Thus, there are 24 classes of 25 students each and one class of 150 students. a. Find the mean size of these 25 classes. b. Find the mean class size from a student's point of view, noting that each student has five classes containing \(25,25,25,25\), and 150 students, respectively. Are the means in parts a and \(\mathrm{b}\) equal? If not, why not?

The following data represent the numbers of tornadoes that touched down during 1950 to 1994 in the 12 states that had the most tornadoes during this period (Storm Prediction Center, 2009). The data for these states are given in the following order: CO, FL, IA, IL, KS, LA, MO, MS, NE, OK, SD, TX. \(\begin{array}{llllllllllll}1113 & 2009 & 1374 & 1137 & 2110 & 1086 & 1166 & 1039 & 1673 & 2300 & 1139 & 5490\end{array}\) a. Calculate the mean and median for these data. b. Identify the outlier in this data set. Drop the outlier and recalculate the mean and median. Which of these two summary measures changes by a larger amount when you drop the outlier? c. Which is the better summary measure for these data, the mean or the median? Explain.
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