Problem 5 - 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. \(z=-0.7\)

Step-by-Step Solution

Answer

24.20% of data items lie below a z-score of -0.7, and 75.80% lie above.

Step 1: Use the Z-Table to find percentage below the given z-score

A z-table is used to find the percentage of values below a particular z-score. For a z-score of -0.7, look it up in the z-table; it shows roughly 0.2420 or 24.20%. Thus, 24.20% of data items are below a z-score of -0.7.

Step 2: Find percentages above the z-score

To find the percentage of data items above the given z-score, subtract the percentage below -0.7 from 100%. This is because the sum of percentages for all data items in a normal distribution is 100%. Hence, 100% - 24.20% = 75.80%.

Key Concepts

z-score

The z-score is a statistical measurement that tells us how many standard deviations a given data point is from the mean of a dataset. In a normal distribution, the mean is the center point from which data points deviate. A z-score can help determine where a particular data item lies in relation to the rest of the data.To calculate a z-score for a specific data point, use the formula:\[ z = \frac{X - \mu}{\sigma} \]Where:

• \( X \) is the data point in question.
• \( \mu \) is the mean of the data set.
• \( \sigma \) is the standard deviation.

A negative z-score, such as -0.7, means the data point is below the mean. The further from zero the score is, the further from the mean it lies. This measurement is crucial when using the z-table to look up probabilities.

z-table

A z-table, or standard normal distribution table, is a tool used to find the area (or probability) below a given z-score on a standard normal distribution curve. It helps in portraying how data values are distributed across a spectrum. Finding a z-score like -0.7 in the z-table, you get the proportion of observations below it. The table is organized so that:

• The left-hand column and top row correspond to the z-scores.
• The body of the table provides the probability or area to the left of that z-score.

For the z-score of -0.7, the table shows a probability of 0.2420, meaning 24.20% of the data or observations are below it. This information can be utilized to derive insights about how your data is spread around the mean in the distribution.

percentage calculation

Percentage calculation in a normal distribution involves determining the proportion of data points either above or below a specific z-score. Once you have a z-score, and you find the percentage below it using the z-table, you can easily compute the percentage of data points above it by subtracting the below percentage from 100%.Here's how you can do it step-by-step:

• Find the percentage of data items below the z-score using the z-table.
• For \( z = -0.7 \), this percentage is 24.20%.
• To find the percentage above the z-score, subtract the below percentage from 100%.
• Thus, 100% - 24.20% = 75.80%.

This process shows that 75.80% of the data items lie above a z-score of -0.7. Understanding percentage calculations is vital in inferential statistics as it sheds light on how data is distributed in practical scenarios.