Problem 64 - Thinking Mathematically

Question

A set of data items is normally distributed with a mean of 400 and a standard deviation of 50. In Exercises \(59-66\), find the data item in this distribution that corresponds to the given z-score. \(z=-2\)

Step-by-Step Solution

Answer

The data item in the distribution that corresponds to z-score -2 is 300.

Step 1: Understand the given values

As provided by the problem parameters, the mean, \(\mu = 400\), standard deviation, \(\sigma = 50\), and the z-score is given as \(z = -2\).

Step 2: Rearrange the z-score formula

The task is to find an original data value, so have to use the formula for a z-score to figure this out. Here, need to find \(x\), so let's rearrange the formula in terms of \(x\), which will look like this: \(x = z \cdot \sigma + \mu\)

Step 3: Substitute the values into the rearranged formula

Substitute the given values: \(x = -2 \cdot 50 + 400\)

Step 4: Solve the equation

Calculating the simplified equation gives the original data value: \(x = -100 + 400 = 300\)

Key Concepts

Normal Distribution

Understanding the concept of a normal distribution is fundamental in statistics. It's a continuous probability distribution that appears as a bell-shaped curve, which visually represents the distribution of data points.

Most of the data points cluster around the central peak, and the probability for a data item decreases as we move away from the mean. This symmetrical distribution implies that the mean, median, and mode of the data set are all equal and lie at the center of the distribution.

Why is this important? The normal distribution is used to determine the probability of a data point occurring within a certain range. When working with any data set that follows this distribution, calculating the probability of specific outcomes becomes a lot easier. Our textbook exercise uses this principle to determine the original data value associated with a given z-score.

Standard Deviation

Standard deviation is a measure that tells us how spread out the numbers are in a data set. It's represented by the Greek letter sigma \( \(\sigma\) \). In essence, if the standard deviation is small, it means the data points are close to the mean, resulting in a steeper and narrower bell curve in a normal distribution graph.

On the other hand, a large standard deviation indicates a wide spread of data points, leading to a flatter bell curve. The exercise provided deals with a standard deviation of 50. This figure helps in understanding how much a data item can theoretically deviate from the mean of 400, before calculating the exact value using the z-score.

Mean in Statistics

The mean, often referred to as the average, is the central point in a data set and is calculated by adding all the data items together and then dividing by the number of items. Statistics use the Greek letter mu \( \(\mu\) \) to represent the mean.

In a normal distribution, the mean serves as the balancing point and is pivotal in calculating z-scores. It is also the point from where we measure the standard deviation. In our exercise, the mean is given as 400, which we use alongside the standard deviation to compute the actual data value corresponding to a specific z-score.