Question

Assume that the distribution of a college entrance exam is normal, with a mean of 500 and a standard deviation of 100 . For each of the following scores, find the equivalent \(Z\) score, the percentage of the area above the score, and the percentage of the area below the score.

Short answer

For a score of 550: Z = 0.5, 69.15% below, 30.85% above.

Step-by-step solution

- Understand the Problem

Determine the equivalent Z-score, the percentage of the area above the score, and the percentage of the area below the score, given a normal distribution with a mean (µ) of 500 and a standard deviation (σ) of 100.

- Z-Score Formula

Use the Z-score formula: \[ Z = \frac{x - \mu}{\sigma} \] - where \(x\) is the given score, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

- Calculate Z-Score

Substitute the values into the formula. For each score provided: i) Let's assume the score is 550: \[ Z = \frac{550 - 500}{100} = \frac{50}{100} = 0.5 \] ii) Repeat for other scores as needed (e.g., 450): \[ Z = \frac{450 - 500}{100} = \frac{-50}{100} = -0.5 \]

- Use Z-Score Table

Find the percentage of the area below the Z-score using the Z-score table. For \(Z = 0.5\), the table shows 0.6915, which means 69.15% of data is below this score. For other scores, look up their values in the Z-score table similarly.

- Calculate Area Above the Z-Score

Subtract the Z-score table value from 1 to find the area above the Z-score. For \(Z = 0.5\): \[ 1 - 0.6915 = 0.3085 \] or 30.85% of the data is above this score. Repeat for other scores using their Z-scores.

- Summarize the Results

Compile all results: For a score of 550: - Z-score: 0.5 - Area below: 69.15% - Area above: 30.85% Repeat similarly for other scores.

Key concepts

Z-score calculation

Let's start by understanding what a Z-score is. A Z-score indicates how many standard deviations a particular score is away from the mean. It is calculated using the formula: \[ Z = \frac{x - \mu}{\sigma} \] where

• \(x\) is the score you're examining.
• \(\mu\) is the mean of the distribution.
• \(\sigma\) is the standard deviation.

For example, if you have a score of 550, a mean of 500, and a standard deviation of 100, you plug these values into the formula: \[ Z = \frac{550 - 500}{100} = \frac{50}{100} = 0.5 \] This result means the score of 550 is 0.5 standard deviations above the mean. Calculating Z-scores for other scores follows the same method. For a score of 450, the Z-score would be: \[ Z = \frac{450 - 500}{100} = \frac{-50}{100} = -0.5 \] This indicates 450 is 0.5 standard deviations below the mean.

Z-score table

After calculating the Z-score, you need to use the Z-score table to find the corresponding percentage of the area under the normal curve. This table provides the cumulative probability from the left up to your Z-score. For example, if you have a Z-score of 0.5, you find 0.5 on the Z-score table which shows a value of 0.6915. This means that 69.15% of the data falls to the left of the score. How do you use another example? If the Z-score is -0.5, the table will give you a value indicating the percentage of data to the left of that score, which is around 30.85%. Using the Z-score table might seem tricky at first, but it provides a visual way to understand where your score lies in the context of the entire data set.

Percentage area under the curve

Understanding the percentage area under the curve is crucial in statistics. This tells us the proportion of data that falls above or below a particular score. Once you have the Z-score and its table value, you can interpret the percentage area under the curve. For example:

• From the Z-score table, if the percentage below a Z-score of 0.5 is 69.15%, this means that 69.15% of the data lies below the score of 550.
• To find the percentage of data above the score, subtract the table value from 1: \[ 1 - 0.6915 = 0.3085 \] This means 30.85% of the data is above the score of 550.

This method allows you to understand the proportion of data falling around specific points on a normal distribution, aiding in various statistical analyses and interpretations.