Problem 54 - 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. 8

Step-by-Step Solution

Answer

The z-score for a score of 8 on the dental anxiety scale is \(-0.75\)

Step 1: Identify the variables.

Here, the value from the dataset (x) is the score on the dental anxiety scale, which is 8. The mean of the dataset (\(\mu\)) is 11, and the standard deviation (\(\sigma\)) is 4.

Step 2: Substitute the values into the z-score formula.

Substitute the values from Step 1 into the z-score formula (\(Z = \frac{x - \mu}{\sigma}\)). The z-score calculation thus becomes \(Z = \frac{8 - 11}{4}\)

Step 3: Calculate the z-score.

After subtracting 11 from 8 and dividing the result by 4, the z-score is computed to be \(-0.75\)

Key Concepts

Understanding the Normal Distribution

Imagine a scenario where students' scores on a test are plotted on a graph. If most students score around the average, with fewer achieving very high or very low scores, the points on the graph would form a shape called the normal distribution, which looks like a bell curve. This shape is symmetrical, with the highest point at the mean. Values spread out from the mean in a predictable pattern: about 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three.

In the case of the dental anxiety scale, the scores range from 0 to 20, and the normal distribution tells us that most people's anxiety levels will cluster around the mean of 11. So, this tells us a lot about how common different levels of dental anxiety are in the population being studied.

Significance of Standard Deviation

The standard deviation is a crucial concept that indicates how spread out the numbers in a set of data are. In the context of the dental anxiety scale, a standard deviation of 4 means that many individuals' scores vary by 4 points above or below the mean score of 11.

To visualize this, picture the bell curve of the normal distribution, with the center at 11. One standard deviation to the right would extend to 15 and one to the left down to 7, accounting for most people's anxiety levels. Standard deviation helps researchers and students alike understand the degree of variance in any given population or set of data, a vital piece of knowledge when analyzing survey results or test scores.

The Importance of Statistical Concepts

Statistical concepts such as the z-score play a major role in interpreting data. A z-score indicates how many standard deviations a data point is from the mean. When a z-score is negative, as in the dental anxiety score of 8 with a z-score of -0.75, it signifies that the value is below the mean. Conversely, a positive z-score would mean the data point is above the mean.

Understanding z-scores helps in determining the rarity or commonality of a data point within a set. For example, a z-score closer to 0 suggests that the data point is very common since it's near the mean. In comparison, a z-score further from 0 indicates a data point that is less common. In educational settings, these statistical concepts enable students to analyze various data sets and comprehend complex research findings in fields ranging from psychology to business.