Problem 51 - 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. 20

Step-by-Step Solution

Answer

The z-score for the score 20 on the dental anxiety scale is 2.25.

Step 1: Identify the given

Given that the mean (\μ) is 11, the standard deviation (σ) is 4, and the score (X) for which we have to find the z score is 20.

Step 2: Plug the given values into the formula

Now, substitute these values in the formula: \(Z = \frac{(X - μ)}{σ}\). It becomes \(Z = \frac{(20 - 11)}{4}\)

Step 3: Solve for the Z-Score

Now, calculate the value in the numerator first, which is \(20-11 = 9\). Then divide this result by the standard deviation, which yields \(Z = \frac{9}{4} = 2.25\)

Key Concepts

Normal Distribution

The normal distribution, often referred to as a bell curve, is a foundational concept in statistics. It represents a distribution where the data is symmetrically distributed around the mean, with most of the observations clustering around the central peak and the probabilities for values further away from the mean tapering off equally in both directions.

The characteristics that define a normal distribution include its symmetrical bell shape and the fact that about 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. This principle is known as the Empirical Rule or the 68-95-99.7 rule.

In the context of the dental anxiety scale problem, the normal distribution tells us that most people's anxiety scores are around the mean of 11, with fewer people having very low (close to 0) or very high (close to 20) anxiety levels.

Standard Deviation

Standard deviation (often abbreviated as SD or \( \sigma \) in formulas) is a measure that quantifies 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, while a high standard deviation indicates that the values are spread out over a broader range.

In the field of statistics, standard deviation is incredibly important for understanding the spread of a data distribution. Calculating the standard deviation involves measuring the difference between each data point and the mean, squaring these differences, averaging them, and finally taking the square root.

Referring to our example, a standard deviation of 4 means that the scores on the dental anxiety scale typically vary by 4 points away from the mean score of 11. It serves as a yardstick for measuring how much a particular score, such as 20, deviates from what is typical or average within the dataset.

Statistical Mean

The statistical mean, often simply called the 'mean', is the average of a set of numbers and is calculated by adding them all together and then dividing by the count of numbers. Represented by the Greek letter \( \mu \) in statistics, it is one of the most basic measures of central tendency, indicating the central or typical value in a distribution of data.

A mean is particularly useful when comparing individual scores to a collective average to determine whether they're above, below, or at the average. However, the mean can be sensitive to extreme scores (outliers), which can pull the value of the mean towards them.

In the dental anxiety scale exercise, the given mean of 11 acts as a benchmark to assess the level of anxiety. For instance, a score of 20 on the scale is considerably higher than the average, indicating a higher level of anxiety compared to the 'typical' person.