Problem 41 - 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. 60

Step-by-Step Solution

Answer

The z-score is 0.

Step 1: Understanding the Z-Score

The z-score is a measurement of how many standard deviations an element is from the mean. It can be calculated using the formula: \(Z = \frac{(X - \mu)}{\sigma}\), where 'X' is the data item, '\(\mu\)' is the mean, and '\(\sigma\)' is the standard deviation.

Step 2: Insert the given values into the Z-score formula

Given that the data item 'X' is 60, the mean '\(\mu\)' is 60 and the standard deviation '\(\sigma\)' is 8, plug these values into the formula: \(Z = \frac{(60 - 60)}{8}\)

Step 3: Simplify and calculate the Z-score

Simplify the equation to: \(Z = \frac{0}{8}\). Solve the equation, and the result is Z = 0. This is the Z-score for the given data item.

Key Concepts

Understanding Normal Distribution

Imagine you're measuring the heights of a large group of people. Most individuals will likely have heights close to the average, with fewer people being very short or very tall. This pattern of distribution, where values cluster around a central point, is what we call normal distribution. It forms a symmetrical, bell-shaped curve when graphed.

The highest point on this curve is the mean, which is the average of all the data. In our exercise, the mean is 60. This mean corresponds to the peak of the curve, where most data points lie. Data points that fall far from the mean are less frequent, so the curve tapers off on either side. This distribution is central to many statistical methods and is a crucial concept in fields ranging from psychology to finance.

Standard Deviation and Variability

In any group of data, not all values are the same; there's always some spread. Standard deviation is a measure of this spread. It tells us how much each data item tends to differ from the mean. A low standard deviation implies that the data points are close to the mean, and a high standard deviation indicates that the data points are more spread out.

In our example, the standard deviation is 8. This means that, on average, each data item varies by plus or minus 8 from the mean of 60. When we're looking at the normal distribution curve, the standard deviation determines the width of the bell shape: the larger the standard deviation, the wider the bell. Knowing the standard deviation helps in understanding the variability within the data set.

Statistical Analysis with Z-Scores

Statistical analysis encompasses methods to describe, infer, and predict outcomes from data. One of these methods is the use of z-scores, which standardize different data sets to be comparable. A z-score indicates how many standard deviations an individual data point is from the mean.

In the resolved exercise, the student is asked to calculate the z-score for a data point. By applying the formula (\(Z = \frac{(X - \text{mean})}{\text{standard deviation}}\)), it standardizes the value, making it possible to understand and compare it within the context of the normal distribution. A z-score of 0, as seen in the exercise, means the data point is exactly at the mean. This statistical tool is essential in fields such as psychology, medicine, and economics for tasks including hypothesis testing, quality control, and risk assessment.