Problem 44 - Thinking Mathematically

Question

A set of data items is normally distributed with a mean of 60 and a standard deviation of 8 . In Exercises 33-48, convert each data item to a z-score. 44

Step-by-Step Solution

Answer

The z-score for the data item 44 is -2.

Step 1: Identify the values

From the problem statement, it is known that the mean (also symbolised by \(\mu\)) is 60, standard deviation (also symbolised by \(\sigma\)) is 8 and the individual item's value is 44.

Step 2: Apply the z-score formula

The z-score is computed using the formula \(z= \frac{X - \mu}{\sigma}\), where \(X\) is the individual item's value. Substituting the provided values into the formula, it becomes \(z= \frac{44 - 60}{8}\).

Step 3: Solve the equation

By carrying out the operation, the z-score becomes \(z = \frac{-16}{8} = -2\)

Key Concepts

Normal Distribution

The normal distribution is a fundamental concept in statistics that describes how a set of data points is expected to be distributed across various values. It is often depicted as a bell-shaped curve, which is symmetric about the mean. The majority of the data points fall close to the mean, with fewer and fewer appearing as you move further away. This type of distribution is important because many natural phenomena follow it, making it a key tool in statistical analysis.
Normal distributions are characterized by two parameters:

• The mean ( mu d) which determines the location of the center of the graph.
• The standard deviation ( sigma d) which defines the width of the curve.

This combination of mean and standard deviation ensures that each normal distribution remains unique and can be used to infer probabilities about data values.

Mean

The mean is the average value of a set of numbers, providing a central point around which the data is distributed. It is found by summing all the individual data points and dividing by the total number of points. This value helps describe the typical value you might expect in a data set.
In the context of a normal distribution, the mean ( mu d) is that point of highest frequency, or the peak, representing where most of the data points congregate. In the given example, the mean of 60 signifies that 60 is the most common value around which other data points scatter. Having a good understanding of the mean is crucial, especially when interpreting data because it is foundational to calculating other statistical measures like variance and standard deviation.

Standard Deviation

Standard deviation is a statistic that measures the amount of variability or spread in a set of data. A small standard deviation means that the data points tend to be close to the mean, while a large standard deviation indicates that data points are spread out over a larger range.

• Standard deviation is symbolized by sigma d and calculated as the square root of the variance.
• It provides insight into the consistency and reliability of the mean as a representative of the data set.

In the exercise, the standard deviation is 8, indicating how much the individual data points can vary from the mean value of 60. Understanding standard deviation is vital for determining how much confidence you can place in the mean when making predictions or analyses.

Z-Score

A z-score is a statistical measure that describes a data point's relation to the mean of a group of points. It is expressed as the number of standard deviations away from the mean.
The formula for calculating a z-score is:
\[ z = \frac{X - \mu}{\sigma} \]
Where:

• \(X\): The value of the data point.
• \(\mu\): The mean.
• \(\sigma\): The standard deviation.

In our specific example, converting a data item with a value of 44 to a z-score involves substituting these values into the formula, resulting in a z-score of -2. This means that 44 is 2 standard deviations below the mean of 60. Calculating z-scores is beneficial in determining how unusual or typical a data point is within a distribution.