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Ch. 2 - Descriptive Statistics
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLarson 8th EditionCh. 2 - Descriptive StatisticsProblema 2.CR.6b
Capitolo 2, Problema 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.

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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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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.
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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.
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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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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

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