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Chapter 2: Problem 131

An exam produced grades with a mean score of 74.2 and a standard deviation of \(11.5 .\) Find the \(z\) -score for each test score \(x\) : a. \(x=54\) b. \(x=68\) c. \(x=79\) d. \(x=93\)

Short Answer

Expert verified
The z-scores for the test scores 54, 68, 79, and 93 are approximately -1.76, -0.54, 0.42, and 1.63 respectively.

Step by step solution

01

Understand the Z-score formula

The z-score is calculated as \( z = \frac{x - 渭}{蟽} \) where: \n- `x` is the value from the dataset (the test score), \n- `渭` is the mean of the dataset (mean test score), and \n- `蟽` is the standard deviation of the dataset (standard deviation of the test scores).
02

Calculate Z-score for x=54

Substitute `x = 54`, `渭 = 74.2`, and `蟽 = 11.5` into the z-score formula to find: \n\( z = \frac{54 - 74.2}{11.5} \).
03

Calculate Z-score for x=68

Substitute `x = 68`, `渭 = 74.2`, and `蟽 = 11.5` into the z-score formula to find: \n\( z = \frac{68 - 74.2}{11.5} \).
04

Calculate Z-score for x=79

Substitute `x = 79`, `渭 = 74.2`, and `蟽 = 11.5` into the z-score formula to find: \n\( z = \frac{79 - 74.2}{11.5} \).
05

Calculate Z-score for x=93

Substitute `x = 93`, `渭 = 74.2`, and `蟽 = 11.5` into the z-score formula to find: \n\( z = \frac{93 - 74.2}{11.5} \).

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Key Concepts

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

Mean and Standard Deviation
Understanding the concepts of mean and standard deviation is crucial when working with data. The mean, often called the average, represents the central point of a dataset. It's calculated by summing all values and dividing by the number of values. For our exercise, the mean is 74.2, indicating the average test score.

The standard deviation, represented as \( \sigma \), measures the spread of data around the mean. It shows how much variation exists. In our example, the standard deviation is 11.5, suggesting how scores deviate from the average.
  • A small standard deviation means data points are close to the mean.
  • A large standard deviation indicates a wide spread of scores.
Statistical Formulas
Statistical formulas are tools used to analyze and interpret data. One important formula in statistics is the z-score formula:

\[ z = \frac{x - \mu}{\sigma} \]
- \( x \) is the individual data point.- \( \mu \) is the mean of the data.- \( \sigma \) is the standard deviation.

This formula calculates how far away a specific score is from the mean in terms of standard deviations. It helps standardize scores by providing a way to compare them directly.
  • The z-score tells us how many standard deviations a value is from the mean.
  • A positive z-score indicates a value above the mean, while a negative score shows it's below.
Test Scores Analysis
Analyzing test scores using z-scores provides insight into how individual scores compare to the average. By calculating the z-score for different values, students can understand their performance relative to peers.

For instance, by applying the z-score formula:
  • For \( x = 54 \), the score is far below the mean, resulting in a negative z-score.
  • \( x = 68 \) still falls below the mean, but closer, giving a smaller negative z-score.
  • \( x = 79 \) is above the mean, resulting in a positive z-score.
  • \( x = 93 \) scores well above the mean, shown by a high positive z-score.
Understanding z-scores helps in identifying outliers or exceptional performers and provides a standardized way to compare different test scores.

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