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Chapter 12: Problem 80

Give an example of both a commonly occurring and an infrequently occurring \(z\)-score. Explain how you arrived at these examples.

Short Answer

Expert verified
An example of a commonly occurring z-score is 0, which represents the mean, and an infrequently occurring z-score would be 2, which represents a value that is two standard deviations away from the mean.

Step by step solution

01

Formula for z-score

The formula for calculating the z-score is: \(z = \frac{{X - \mu}}{{\sigma}}\), where \(X\) is the data point, \(\mu\) is the mean of the data and \(\sigma\) is the standard deviation of the data.
02

Define dataset

Let's consider a dataset where student's ages in a class occur. The ages are in the range of 14 to 19. Assume that the mean (\(\mu\)) age is 16 and the standard deviation (\(\sigma\)) is 1.5.
03

Calculate commonly occurring z-score

A commonly occurring z-score would be the age 16 itself, as it's the mean. So, \(X = 16\). Substituting these values into the formula we get: \(z = \frac{{16 - 16}}{{1.5}} = 0\).
04

Calculate infrequently occurring z-score

An infrequent z-score could be for the age 19, as it's well beyond the mean. So, \(X = 19\). Substituting these values into the formula we get: \(z = \frac{{19 - 16}}{{1.5}} = 2\).

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Most popular questions from this chapter

Compute the mean, range, and standard deviation for the data items in each of the three samples. Then describe one way in which the samples are alike and one way in which they are different. Sample A: \(8,10,12,14,16,18,20\) Sample B: \(8,9,10,14,18,19,20\) Sample C: \(8,8,8,14,20,20,20\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. My math teacher gave a very difficult exam for which the distribution of scores was skewed to the right.

Describe the normal distribution and discuss some of its properties.

Intelligence quotients (IQs) on the Stanford-Binet intelligence test are normally distributed with a mean of 100 and a standard deviation of 16. Use the 68-95-99.7 Rule to find the percentage of people with IQs below \(84 .\)

Find the standard deviation for each group of data items. Round answers to two decimal places. \(1,1,1,4,7,7,7\)
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