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Chapter 22: Problem 10

Use the following information. A standardized math test has a mean score of 200 and a standard deviation of \(15 .\) Find and interpret the z-scores of the following math test scores. $$179$$

Short Answer

Expert verified
The z-score is \(-1.4\), meaning the score is 1.4 standard deviations below the mean.

Step by step solution

01

Understand the Problem

We need to find the z-score for a test score of 179. The test has a mean (average) score of 200 and a standard deviation of 15.
02

Recall the Z-Score Formula

The formula to calculate a z-score is: \[ z = \frac{(X - \mu)}{\sigma} \] Where \( X \) is the test score, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
03

Substitute the Values into the Formula

Substitute the given values into the z-score formula: \[ z = \frac{(179 - 200)}{15} \]
04

Perform the Calculation

Calculate the numerator: \( 179 - 200 = -21 \) Then, divide by the standard deviation: \[ z = \frac{-21}{15} \] So, \( z \approx -1.4 \).
05

Interpret the Z-Score

A z-score of \(-1.4\) indicates that the score of 179 is 1.4 standard deviations below the mean score of 200.

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Key Concepts

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

Understanding Standard Deviation
Standard deviation is a concept that measures the amount of variation or dispersion in a set of values. When dealing with a standardized test, the standard deviation tells us how much the test scores are spread out from the mean score.
If the standard deviation is small, it means that most of the test scores are close to the mean. Conversely, a larger standard deviation indicates that the scores are more spread out.
  • In the context of the given problem, the test has a standard deviation of 15. This figure suggests there is a moderate spread in test scores around the mean of 200.
Understanding standard deviation is key to interpreting z-scores, as it gives context to how individual scores compare to the overall group.
Mean Score Simplified
The mean score is essentially the average score obtained by students in a test. Calculating the mean involves adding up all the test scores and then dividing by the number of scores.
In standardized tests, the mean provides a benchmark to evaluate individual performance. This average helps identify which scores are typical and which are outliers.
  • For this exercise, the standardized math test has a mean score of 200, which serves as the central point in the distribution of scores.
Understanding the mean is crucial because it is used as the reference point in the z-score formula.
Introduction to Standardized Tests
A standardized test is designed in such a way that it is consistent and fair for all test takers. These types of tests are often used in educational settings to measure a student's performance against a common standard or benchmark.
The concept of a standardized test is that every participant answers the same set of questions under the same conditions, which allows for comparisons across different test-takers.
  • Since these tests usually report scores in terms of how far above or below the average a particular score lies, interpreting scores using the mean and standard deviation is essential.
This background information helps in understanding why z-scores are used to analyze performance on standardized tests.
Explaining Z-Score Interpretation
The z-score represents the number of standard deviations a data point is from the mean. It's a way of understanding how unusual or typical a score is within a distribution.
To calculate a z-score, you use the formula: \[ z = \frac{(X - \mu)}{\sigma} \] where \( X \) is the score you are examining, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
  • In the given exercise, the z-score of -1.4 for a test score of 179 means that this score is 1.4 standard deviations below the mean of 200.
  • This interpretation allows us to determine that a score of 179 is relatively lower than the average score of the test takers, helping educators and students understand performance in relation to the group.
Understanding z-scores is fundamental in assessing whether a score is above, below, or at the average compared to others.

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Most popular questions from this chapter

Use the following data. In a random sample. 500 college students were asked which social networks they use on a daily basis. The results are summarized below: $$\begin{array}{l|l} \text {Social Network} & \text {Frequency} \\ \hline \text { Facebook } & 305 \\ \text { Instagram } & 255 \\ \text { Twitter } & 175 \\ \text { Google }+ & 115 \\ \text { Pinterest } & 80 \\ \text { Vine } & 80 \end{array}$$ Make a bar graph of these data showing relative frequency on the vertical axis.

Use the following data. In a random sample. 500 college students were asked which social networks they use on a daily basis. The results are summarized below: $$\begin{array}{l|l} \text {Social Network} & \text {Frequency} \\ \hline \text { Facebook } & 305 \\ \text { Instagram } & 255 \\ \text { Twitter } & 175 \\ \text { Google }+ & 115 \\ \text { Pinterest } & 80 \\ \text { Vine } & 80 \end{array}$$ Make a bar graph of these data showing frequency on the vertical axis.

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Use the following information. If the weights of cement bags are normally distributed with a mean of 60 lb and a standard deviation of 1 lb, use the empirical rule to find the percent of the bags that weigh the following: Between 58 Ib and 62 lb
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