Problem 7 - Thinking Mathematically

Question

In Exercises 1-8, find the percentage of data items in a normal distribution that lie a. below and b. above the given z-score. \(z=-1.2\)

Step-by-Step Solution

Answer

a. Approximately 11.51% of the data items are below \(z = -1.2\). b. Approximately 88.49% of the data items are above \(z = -1.2\).

Step 1: Identify the given z-score

The z-score given in the exercise is \(z = -1.2\).

Step 2: Look up the value in a z-score table

Using a z-score table, look up the value corresponding to \(z = -1.2\). This value represents the area to the left (below) of \(z = -1.2\) on the standard normal distribution. In standard z-tables, this value is approximately 0.1151 or 11.51%.

Step 3: Calculate the percentage above the given z-score

The area to the left of \(z = -1.2\) represents the percentage of data items below the z-score. Because the total area under the curve (representing 100% of data items) is 1 (or 100%), subtracting the area to the left of the z-score from 1 will give the area to the right of the z-score, or the percentage of data items above the given z-score. So, \(1 - 0.1151 = 0.8849\) or 88.49%.

Key Concepts

Normal Distribution

The normal distribution, often called the bell curve due to its shape, is a fundamental concept in statistics representing how data tends to spread around a mean (or average) value. Most values cluster around a central region and the probabilities for taking on values further away from the mean taper off equally in both directions.

Understanding normal distribution is vital as it applies to various natural phenomena and measurement errors. This symmetrical distribution is defined by the mean (the peak) and the standard deviation (how spread out the values are). When plotted on a graph, the normal distribution will show most of the data within one standard deviation of the mean and progressively fewer and fewer as you move away from the mean.

Standard Normal Distribution

Z-Score and Standard Normal Distribution

When we standardize a normal distribution, we create a 'standard normal distribution', which has a mean of 0 and standard deviation of 1. This standardization process involves converting scores, called raw scores, into z-scores. A z-score indicates how many standard deviations an element is from the mean.

With z-scores, we can compare data from different normal distributions since z-scores are standardized. They also allow us to calculate the probability of a score occurring within a normal distribution, and these probabilities are the same for all standard normal distributions. This universality makes the standard normal distribution especially valuable in statistics.

Z-Score Table

Using a Z-Score Table

A z-score table, also known as a standard normal table, is a reference table that provides the probability of a random variable falling below a particular value in a standard normal distribution. It is a powerful tool for statisticians to determine the percentage of values that lie above or below a certain point.

Generally, these tables provide the area to the left of a z-score. For example, if we have a z-score of -1.2, the table gives us the area to the left of this score. Taking the area from the table (in our example, 11.51%) gives us the percentage of data below the z-score. To find the percentage above, you'd subtract this value from 100%. It's essential to become comfortable reading these tables accurately to perform statistical computations.

Percentage Calculation in Statistics

Calculating Percentages in a Distribution

Statistics often involve calculating the percentage or probability of data being within a certain range. This is achieved by integrating the area under the curve of a distribution, which determines the likelihood of a variable taking on a specified value.

In normal distributions, this is done using the standard normal distribution and z-scores. After finding the respective area under the curve (either from a z-score table or using software), we express the cumulative area as a percentage of the total area (100%). The result is the percentage of observations lying below or above a specific value, allowing us to draw conclusions about the data set we're analyzing.