Problem 56 - 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. 1

Step-by-Step Solution

Answer

-2.5

Step 1: Identify the Score, Mean, and Standard Deviation

Here the given score (X) is 1. The provided mean (µ) is 11 and the standard deviation (σ) is 4.

Step 2: Insert the Variables into the Formula

Now, plug in the values from Step 1 into the z-score formula: z = (X - µ) / σ. Thus, Z stands for z-score, X=1, µ=11 and σ=4.

Step 3: Perform the Calculations

Substitute the above obtained values into the z-score formula. So, z = (1 - 11) / 4 = -10 / 4 = -2.5

Key Concepts

Understanding Normal Distribution

Imagine a smooth, symmetrical, bell-shaped curve. That’s a visual representation of a normal distribution. In statistics, data often clusters around a central point and spreads out evenly on both sides, forming this characteristic shape.
The dental anxiety scores in our example are distributed normally, which means most people score around the average, with fewer people scoring very high or very low. A normal distribution is useful because it allows us to predict how likely it is for a data point to fall within a certain range.

• The highest point of the curve is at the mean.
• The spread or width of the curve is determined by standard deviation.
• Data follows a predictable pattern, meaning approximately 68% of values lie within one standard deviation from the mean.

Knowing how data is distributed helps in calculating z-scores and understanding other statistical concepts.

Mean and Standard Deviation in Context

The mean and standard deviation are crucial in study of data spread.
Here, the mean represents the average dental anxiety score, which is 11. This is the central value of all the scores compiled in the study.
Mean tells us where the center of the data lies, making it a critical reference point.

• The mean is like the balance point of the data.
• All scores cluster around this point but spread differently based on the variability.

Now, let's talk about standard deviation, which in this case is 4. This statistic measures how spread out the scores are around the mean.
In simple words, it tells you how "varied" the data is.
If the standard deviation is small, the scores are close to the mean; if it's large, they are more spread out. Understanding these two measures helps in calculating the z-score effectively.

The Dental Anxiety Scale

The Dental Anxiety Scale is a specific tool used to measure a person's anxiety associated with dental visits. This scale ranges from 0 to 20, where 0 indicates no anxiety, and 20 indicates extreme anxiety.
In this exercise, the scale is used to understand the anxiety distribution among different individuals.

• It helps identify individuals who might require additional support or interventions.
• Scales like this one are often used in psychological studies to quantify qualitative experiences.

Understanding the use of such scales allows for better psychological profiling and tailored treatment plans. In our exercise, by analyzing scores with the normal distribution, we can assess how typical or atypical a specific score, such as 1, is within the population.