Problem 48 - 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. 30

Step-by-Step Solution

Answer

The z-score for the given data value is -3.75

Step 1: Write down the known values from the problem

From the problem, we know that the Mean (μ) = 60, Standard Deviation (σ) = 8, and the individual data point (X) = 30.

Step 2: Apply the z-score formula

Let's apply the formula for z-score. \(Z = (X - μ) / σ\) or \(Z = (30 - 60) / 8\)

Step 3: Simplify the expression

Simplify the expression, \(Z = -30 / 8\)

Step 4: Solve the division

Solving the division we get \(Z = -3.75\)

Key Concepts

Standard Deviation

The standard deviation is a measure that quantifies the amount of variation or dispersion of a set of data values. A low standard deviation indicates that the data points tend to be close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range. In the context of the exercise, a standard deviation of 8 means that, on average, each data point is 8 units away from the mean.

Understanding standard deviation is crucial as it is used in the calculation of the z-score, which measures the number of standard deviations a data point is from the mean. This can be visualized by imagining a bell curve: data points close to the mean fall near the top-center of the curve, and those further away towards either end.

Normal Distribution

A normal distribution, also known as the bell curve, 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. In a normal distribution, about 68% of data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and about 99.7% falls within three standard deviations.

Our exercise indicates that the data is normally distributed with a mean of 60. Since the data item in question (30) is much lower than the mean and beyond one standard deviation, it is in the lower tail of the distribution.

Statistical Methods

Statistical methods are a set of tools used to collect, analyze, present, and interpret data. They are essential in research, enabling us to make sense of the data collected and to draw meaningful conclusions. In our exercise, we utilize one of these methods, the z-score calculation, which helps us determine how unusual or typical a value is within a data set.

The z-score is particularly useful in fields like psychology, finance, and other sciences where comparing different data sets with different units or different variances is necessary. These standardized scores allow comparisons across different data sets and are foundational in understanding the normal distribution.

Data Analysis

Data analysis involves inspecting, cleaning, transforming, and modeling data with the goal of discovering useful information, drawing conclusions, and supporting decision-making. Understanding how to calculate the standard deviation and z-scores is part of exploratory data analysis, which allows for an understanding of the properties of the data.

In the given exercise, we use data analysis to convert raw scores into standardized z-scores, thereby allowing us to see how many standard deviations away from the mean a particular score is. This z-score conversion is central to many statistical analyses, especially when it comes to understanding the relationship between data points in a normal distribution.