Problem 49 - Thinking Mathematically

Question

Scores on a dental anxiety scale range from 0 (no anxiety) to 20 (extreme anxiety). The scores are normally distributed with a mean of 11 and a standard deviation of 4. In Exercises 49-56, find the z-score for the given score on this dental anxiety scale. 17

Step-by-Step Solution

Answer

The z-score for the given score of 17 on the dental anxiety scale is 1.5.

Step 1: Set Up the Given Information

First, identify and record the given information. In this case, we know that the given score (X) is 17, the mean (μ) is 11, and the standard deviation (σ) is 4.

Step 2: Apply the Z-score Formula

Insert the given values into the z-score formula, Z = (X - μ)/ σ. Substituting for X, μ and σ, we get Z = (17 - 11)/ 4.

Step 3: Perform the Calculation

Perform the subtraction and division to calculate the z-score. In this case, 17 - 11 gives 6 and 6 divided by 4 gives a z-score of 1.5.

Key Concepts

Normal Distribution

A normal distribution is a key conceptual foundation in statistics that graphically represents the distribution of a set of data points. Imagine a bell-shaped curve where the bulk of the values lie around the mean, or average, with fewer and fewer values falling farther away from the mean. This curve is symmetric, meaning that the right side is a mirror image of the left side. In the context of our dental anxiety scale example, the distribution of test scores would form a bell-shaped curve around the mean score of 11. Most individuals' anxiety scores will hover near this average, with fewer people reporting very low or very high anxiety.

In practice, this normal distribution means that we can predict probabilities and understand the likelihood of certain outcomes. For example, we know that in a normal distribution, about 68% of the observed data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three. This becomes incredibly useful when analyzing data and making informed decisions based on statistical patterns.

Standard Deviation

The concept of standard deviation is integral to understanding how spread out a group of numbers is in a data set. It tells us the amount of variability or spread in a distribution of data. A lower standard deviation indicates that the data points tend to be closer to the mean, while a higher standard deviation indicates that the data points are spread out over a wider range of values.

In the dental anxiety score example, the standard deviation is given as 4. This means that the scores of most individuals are expected to fall within a range of 4 points above or below the mean score of 11. It's a way to quantify uncertainty. When we calculate the z-score for a particular value, like a score of 17, we're effectively measuring how many standard deviations that value is away from the mean. This helps us understand how unusual or common a particular data point is within the context of the entire data set.

Statistical Analysis

Statistical analysis involves collecting, exploring, and presenting large amounts of data to discover underlying patterns and trends. It's essentially about making sense of data by using various methods and tools, one of which is the calculation of z-scores. This calculation is frequently used in statistical analysis because it allows comparing different data points within a distribution, regardless of the scale of the original values.

Performing a z-score calculation in our dental anxiety scale scenario provides a clear example of such analysis in action. By converting the raw score of 17 into a z-score, we can precisely determine how this score compares to the rest of the data. We learn that it is 1.5 standard deviations above the mean, which provides us with a standardized way to analyze and interpret this specific score in the context of the normally distributed data we're working with. In broader terms, statistical analysis empowers us to make educated decisions and to understand the significance and implications of data in research, business, and many other fields.