Problem 10 - Thinking Mathematically

Question

In Exercises 9-16, find the percentage of data items in a normal distribution that lie between \(z=0.3\) and \(z=2.1\).

Step-by-Step Solution

Answer

The percentage of data items that lie between \(z=0.3\) and \(z=2.1\) is approximately 36.42%.

Step 1: Identify the z-scores

Identify the given \(z\)-scores. Here, the \(z\)-scores are \(z_1 = 0.3\) and \(z_2 = 2.1\). The task is to find the percentage of data that lies between these two \(z\)-scores in a standard normal distribution.

Step 2: Find the area related to \(z_1\)

The area to the left of \(z_1 = 0.3\) can be found using a Z-table or a calculator with a normal distribution function. The value, from looking up \(z = 0.3\) in the Z-table, is approximately 0.6179 or 61.79%. This value means that approximately 61.79% of the data is less than \(z = 0.3\).

Step 3: Find the area related to \(z_2\)

The same process is applied to \(z_2 = 2.1\). Looking up \(z = 2.1\) in the Z-table yields a value is approximately 0.9821 or 98.21%. This value means that approximately 98.21% of the data is less than \(z = 2.1\).

Step 4: Get the percentage of data

To get the percentage of data between \(z_1\) and \(z_2\), subtract the area related to \(z_1\) from the area related to \(z_2\). Hence, 0.9821 - 0.6179 = 0.3642. This is equivalent to \(36.42\%\). So, approximately \(36.42\%\) of the data items are between \(z = 0.3\) and \(z = 2.1\).

Key Concepts

Understanding Z-scores

Z-scores are a way of standardizing scores on the same scale by calculating the number of standard deviations a particular data point is from the mean of the dataset. When you have a normal distribution, z-scores help to determine where exactly a data point lies in comparison to the entire distribution. It's like finding your position in a queue—you know exactly how far or close you are to the front. When you convert a score to a z-score, you are literally transforming the data point into a relative position—

• If the z-score is 0, it means the data point is exactly at the mean.
• A positive z-score indicates a data point above the mean, while a negative score indicates a point below it.

This transformation allows for easy comparison and calculation across different datasets or distributions.

How to Use the Z-table

The Z-table is a crucial tool for anyone dealing with normal distributions. It contains probabilities that correspond to specific z-scores, giving you the area under the curve to the left of that z-score.
Here's how to use it:

• Find the z-score you are interested in. Make sure to note both the whole number and precise decimal places for correct lookup.
• Find the corresponding row in the Z-table that matches the first digits of your z-score.
• Move along that row to find the column matching the second decimal place of the z-score.

For example, if you have a z-score of 0.3, you check row 0.3, and the number under column 0.00. The number you find is the cumulative probability, which tells you the percentage of data below this z-score.

Exploring the Standard Normal Distribution

The standard normal distribution is a type of normal distribution with a mean of 0 and a standard deviation of 1. All normal distributions can be transformed into a standard normal distribution by converting individual scores into z-scores. This serves as a universal reference or model for comparing different datasets. It's like having a global unit (like meters) for distance comparison.
The distribution curve is smooth and symmetrical, resembling a bell. Most data points cluster around the mean, creating the peak of the bell, and the probabilities of extreme values decrease as you move away from the mean. This characteristic makes it particularly useful for statistical analysis and probability prediction. It doesn't matter the original mean or standard deviation; once in z-score form, you can apply the universal standard represented by the standard normal distribution.

Calculating Data Percentages

Calculating data percentages in a normal distribution using z-scores allows you to understand the proportion of observations between two scores. Here's a simple approach:

• First, find the cumulative percentage for each z-score using the Z-table.
• Subtract the smaller from the larger cumulative percentage to find the interval percentage.

For instance, in our exercise, the percentage for z = 0.3 is approximately 61.79%, and for z = 2.1 is 98.21%. Subtracting gives you about 36.42%, meaning approximately 36.42% of data falls between those z-scores.
This method is deeply rooted in z-score use, enabling seamless transition from individual scores to understanding their relative frequency within a dataset.