Problem 3 - 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

The percentage of data items below the z-score of 1.2 is 88.49%, and the percentage of data items above the z-score of 1.2 is 11.51%.

Step 1: Understanding normal distribution and z-score

The normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It indicates how many standard deviations an element is from the mean. In this case, the z-score is \(z=1.2\)

Step 2: Find the percentage below the given z-score

The z-table or standard normal distribution table can be used to find the percentage of data below the given z-score. The z-table shows the percentage of data below each z-score, up to two decimal places. If \(z=1.2\), our task is to find the z-table value for 1.2, which is 0.8849 or 88.49%.

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

To find the percentage of data above a given z-score, subtract the z-table value from 1. Here it is \(1 - 0.8849 = 0.1151\) or 11.51%.

Key Concepts

z-score

A z-score is a numerical measurement that tells you how many standard deviations a data point is from the mean. It's a way to standardize data on a uniform scale.
Imagine you have a dataset, and you want to find how each data point compares to the rest. The z-score gives you this insight by showing how far away a value is from the average, in terms of standard deviations.
For example, if a z-score is 1.2, like in our exercise, it indicates the value is 1.2 standard deviations above the mean. z-scores can be positive or negative:

• A positive z-score means the value is above average.
• A negative z-score means it is below average.
• A z-score of 0 would mean the value is exactly at the mean.

z-table

The z-table, also known as the standard normal distribution table, is a mathematical table that helps in finding the percentage of values below a specific z-score in a standard normal distribution.
It's used to determine the probability that a statistic is observed below, at, or above a certain standard deviation from the mean.
Here's how it works: assuming you have a z-score, like 1.2, you find this score in the z-table to learn what percentage of the data falls below that score (in our example, it's 88.49%).
They are handy tools because they save you from performing complex calculations every time, providing a direct way to see the percentile rank of a particular z-score. Remember that most z-tables usually contain cumulative probabilities.

standard deviations

Standard deviations measure the amount of variation or dispersion in a dataset. It tells us how spread out the numbers are in a dataset.
If the data points are close to the mean, the standard deviation will be small. However, if the data points are spread out over a large range of values, the standard deviation will be large.

• Useful for understanding the dispersion around the mean.
• Helps determine how significant a particular z-score is.

Understanding standard deviations is crucial since they are part of the z-score formula. A higher standard deviation means more spread out data, affecting how you interpret z-scores.

Gaussian distribution

The Gaussian distribution, or normal distribution, is one of the most common probability distributions applied in statistics. It describes how data points are distributed around a mean in a symmetrical bell-shaped curve.
Its importance lies in its universal properties: many natural phenomena and measurement errors tend to cluster around a mean, making this distribution a powerful tool.
Key features include:

• The mean, median, and mode are located at the center of the distribution.
• The curve is symmetric about the mean.
• 68% of the data falls within one standard deviation of the mean, approximately 95% within two, and 99.7% within three (Empirical Rule).

Understanding the Gaussian distribution helps make sense of statistical analysis and predicting probabilities, which is often what you'd be doing when working with z-scores and z-tables.