Ch. 2 - Descriptive Statistics

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 2 - Descriptive Statistics

Problem 2.CR.6b

Chapter 2, Problem 2.CR.6b

The mean annual salary for a sample of electrical engineers is \$86,500, with a standard deviation of $1500. The data set has a bell-shaped distribution.

b. The salaries of three randomly selected electrical engineers are $93,500, \$85,600, and \$82,750. Find the z-score that corresponds to each salary. Determine whether any of these salaries are unusual.

Verified step by step guidance

Step 1: Recall the formula for calculating the z-score:

z = (x - μ) / σ

, where

x

is the data point,

μ

is the mean, and

σ

is the standard deviation.

Step 2: Substitute the given values into the formula for each salary. For the first salary, \(93,500, calculate

z = (93500 - 86500) / 1500

. For the second salary, \)85,600, calculate

z = (85600 - 86500) / 1500

. For the third salary, \$82,750, calculate

z = (82750 - 86500) / 1500

Step 3: Simplify the numerator for each calculation. For the first salary, compute

93500 - 86500

. For the second salary, compute

85600 - 86500

. For the third salary, compute

82750 - 86500

Step 4: Divide the result of each numerator by the standard deviation,

1500

, to find the z-scores for each salary.

Step 5: Determine whether any of the z-scores are unusual. Recall that a z-score is considered unusual if it is less than -2 or greater than 2. Compare each calculated z-score to this threshold to identify any unusual salaries.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Z-Score

A Z-score measures how many standard deviations a data point is from the mean of a dataset. It is calculated using the formula Z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation. A Z-score helps in identifying how unusual or typical a value is within a distribution, particularly in a normal distribution.

Standard Deviation

Standard deviation quantifies the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range. In this context, the standard deviation of $1500 provides insight into the variability of electrical engineers' salaries around the mean of $86,500.

Normal Distribution

A normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve. In this scenario, the mention of a bell-shaped distribution implies that the salaries of electrical engineers follow this pattern, allowing for the application of Z-scores to assess the unusualness of specific salaries.

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Related Practice

Textbook Question

Extending Concepts

Midquartile Another measure of position is called the midquartile. You can find the midquartile of a data set by using the formula below.

Midquartile = (Q₁ + Q₃) / 2

In Exercises 55 and 56, find the midquartile of the data set.

5 7 1 2 3 10 8 7 5 3

Textbook Question

Graphing Data Sets In Exercises 17–32, organize the data using the indicated type of graph. Describe any patterns.

Engineering Degrees Use a time series chart to display the data shown in the table. The data represent the number of bachelor’s degrees in engineering (in thousands) conferred in the U.S. (Source: U.S. Deapartment of Education)

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Textbook Question

You are a member of your local apartment association. The association represents rental housing owners and managers who operate residential rental property throughout the greater metropolitan area. Recently, the association has received several complaints from tenants in a particular area of the city who feel that their monthly rental fees are much higher compared to other parts of the city.

You want to investigate the rental fees. You gather the data shown in the table at the right. Area A represents the area of the city where tenants are unhappy about their monthly rents. The data represent the monthly rents paid by a random sample of tenants in Area A and three other areas of similar size. Assume all the apartments represented are approximately the same size with the same amenities.

a. What type of graph would you choose to display the data? Explain your reasoning.

Textbook Question

Tail lengths (in feet) for a sample of American alligators are listed.

6.5 3.4 4.2 7.1 5.4 6.8 7.5 3.9 4.6

a. Find the mean, median, and mode of the tail lengths. Which best describes a typical American alligator tail length? Explain your reasoning.

Textbook Question

c. Based on your data displays, does it appear that the monthly rents in Area A are higher than the rents in the other areas of the city? Explain.

Textbook Question

Weekly salaries (in dollars) for a sample of construction workers are listed.

1100 720 1384 1124 1255 976 718 1316

749 1062 1248 891 969 790 860 1100

a. Find the mean, median, and mode of the salaries. Which best describes a typical salary?