Problem 63 - 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. \(z=-3\)

Step-by-Step Solution

Answer

The data item corresponding to the z-score of -3 is 250.

Step 1: Understand the z-score formula

Given z-score formula is \(z = \frac{x - \mu}{\sigma}\), where \(x\) is the data item we need to find, \(\mu\) is the mean and \(\sigma\) is the standard deviation. In this situation, the mean is 400 and standard deviation is 50. And the given z-score is -3.

Step 2: Rearrange the formula to find x

We need to isolate \(x\) in the formula. To do this, multiply both sides of the equation by \(\sigma\), then add \(\mu\) to both sides, leading to the equation \(x = \sigma*z + \mu\).

Step 3: Use the values of z, \(\sigma\), and \(\mu\) to calculate x

The given z-score is -3, the standard deviation (\(\sigma\)) is 50, and the mean (\(\mu\)) is 400. Substitute these values into the formula: \(x = 50*(-3) + 400 = -150 + 400 = 250\).

Key Concepts

Normal Distribution

Understanding the concept of normal distribution is crucial for analyzing data in many fields, from psychology to education. A normal distribution, commonly known as a bell curve, is a probability distribution that is symmetric around its mean, showing that data near the mean are more frequent in occurrence than data far from the mean. Imagine a bell-shaped curve where the bulk of the points lies in the middle of the axis, tapering off equally towards both ends.

In a normal distribution, roughly 68% of values fall within one standard deviation from the mean, 95% within two standard deviations, and 99.7% within three standard deviations. This distribution is determined by two parameters: the mean and the standard deviation, which explain why it is essential to understand these concepts when dealing with z-scores and other statistical analyses.

Standard Deviation

The term 'standard deviation' often pops up in statistics, and rightly so, as it's a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range.

In practical terms, if you're looking at a set of test scores, a high standard deviation would mean that the students' grades vary widely from the average score. Conversely, a low standard deviation would suggest that most students scored around the classroom average. It's a vital statistic that gives you a rapid sense of the spread of a dataset.

Statistical Mean

The statistical mean might seem like an elementary concept, but it is a cornerstone of statistical analysis. Essentially, the mean is what many would colloquially call the 'average.' It is calculated by adding all the numbers in a dataset together and then dividing by the count of those numbers.

For example, if you have five apples weighing 120, 150, 160, 140, and 130 grams respectively, the mean weight of an apple would be calculated as \( (120 + 150 + 160 + 140 + 130) \div 5 = 140 \) grams. The mean is a form of measure of central tendency and is used as the reference point for the z-score formula, thus highlighting its relevance in contextualizing data.

Z-Score Formula

Navigating through the realm of statistics, the z-score serves as a navigator, providing us the ability to determine how a single data point relates to the rest of a dataset. The z-score formula, \( z = \frac{x - \mu}{\sigma} \) is the standardized value that represents the number of standard deviations a data point, \( x \) is from the mean, \( \mu \).

In simple terms, the z-score tells us whether a data point is above or below the mean, and by how many standard deviations. A z-score of 0 indicates that the data point is exactly at the mean. A positive z-score indicates a value above the mean, while a negative z-score signifies a value below the mean. It's a powerful tool for comparing different datasets and for understanding the relative position of a data point within its distribution.