Problem 40 - Thinking Mathematically

Question

A set of data items is normally distributed with a mean of 60 and a standard deviation of 8 . In Exercises 33-48, convert each data item to a z-score. 78

Step-by-Step Solution

Answer

The z-score for the data item 78 is 2.25.

Step 1: Identify the Necessary Values

Extract the necessary values from the given exercise. The data item \( x \) is 78, the mean \( \mu \) is 60, and the standard deviation \( \sigma \) is 8.

Step 2: Insert the Values Into the Formula

Substitute the given values into the z-score formula \( z = \frac{x - \mu}{\sigma} \). Hence, \( z = \frac{78 - 60}{8} \).

Step 3: Perform the Calculations

First, subtract the mean from the data item \(78 - 60 = 18 \), then divide the difference by the standard deviation \( \frac{18}{8} = 2.25 \).

Key Concepts

Understanding Normal Distribution

At the heart of many statistical analyses lies the normal distribution, often called the bell curve for its distinctive shape. It's a continuous probability distribution that is symmetrical around its mean, showing that data near the mean are more frequent in occurrence than data far from the mean.

In practical terms, if we were to take an infinite number of samples and calculate their means, those means would be normally distributed. For example, heights of adult males, blood pressure measurements, or SAT scores often follow a normal distribution. Recognizing a dataset as normally distributed is integral because it allows the use of several statistical tools and techniques, one of which is the z-score.

Deciphering Standard Deviation

Standard deviation is a critical statistical concept indicating the amount of variation or dispersion present in a set of data values. A low standard deviation suggests that the data points tend to be close to the mean of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values.

In our example, the standard deviation is 8, which means that the majority of data points (about 68% if the data is normally distributed) will be within 8 units above or below the mean. This measure helps us understand how spread out the data is and is crucial when calculating z-scores, as it normalizes the distance of a data point from the mean.

Essential Statistical Concepts

Statistics is laden with concepts that help in summarizing and interpreting data. Apart from normal distribution and standard deviation, some of these include the mean, median, mode, range, and percentiles.

Each serves a unique purpose and it’s important to grasp their implications and uses. For instance, while the mean gives us the average, the median provides a mid-point, and the mode indicates the most frequently occurring value. Statistical concepts are like tools in a toolkit, with each selected to best explain and analyze the characteristics of the data in question.

Grasping the Concept of Mean

The mean, commonly referred to as the average, is calculated by adding up all the numbers in a dataset and dividing by the number of numbers. It is a measure of central tendency, providing a quick glimpse of the 'center' of a data set.

The mean is sensitive to outliers, which are values significantly higher or lower than the rest of the data. In the context of our exercise where the mean is 60, this value represents the center point around which the data items in our normally distributed dataset are clustered. Understanding the mean is crucial when interpreting the z-score, as it is the reference point from which we measure the number of standard deviations a particular data point lies.