Question

Scores on a quiz were normally distributed and had a mean of 10 and a standard deviation of \(3 .\) For each of the following scores, find the \(Z\) score and the percentage of area above and below the score.

Short answer

For X=8: Z=-0.67, 25.14% left, 74.86% right. For X=12: Z=0.67, 74.86% left, 25.14% right. For X=15: Z=1.67, 95.25% left, 4.75% right.

Step-by-step solution

- Understand the Z-score formula

The Z-score formula is given by \[ Z = \frac{X - \mu}{\sigma} \] where:- \(X\) is the score- \(\mu\) is the mean score- \(\sigma\) is the standard deviation

- Calculate Z-scores

Substitute the values into the Z-score formula.For each score \(X\):(a) If \(X = 8\)\[ Z = \frac{8 - 10}{3} = \frac{-2}{3} \approx -0.67 \](b) If \(X = 12\)\[ Z = \frac{12 - 10}{3} = \frac{2}{3} \approx 0.67 \](c) If \(X = 15\)\[ Z = \frac{15 - 10}{3} = \frac{5}{3} \approx 1.67 \]

- Find the percentage area

Use the standard normal distribution table to find the area under the curve corresponding to each Z-score.(a) For \(Z \approx -0.67\): Area to left \( \approx 0.2514 \). Area to right \( \approx 0.7486 \).(b) For \(Z \approx 0.67\): Area to left \( \approx 0.7486 \). Area to right \( \approx 0.2514 \).(c) For \(Z \approx 1.67\): Area to left \( \approx 0.9525 \). Area to right \( \approx 0.0475 \).

Key concepts

normal distribution

A normal distribution, often referred to as a bell curve, is a probability distribution that is symmetric about the mean. This means most of the data points cluster around the average value, with fewer points appearing as you move further away from the mean. In simpler terms, in a normal distribution:

• The mean, median, and mode of the distribution are all equal and located at the center of the distribution.
• The distribution is perfectly symmetrical about the mean.
• Nearly all of the data (about 99.7%) lies within three standard deviations from the mean.

This property makes the normal distribution extremely useful in statistics for understanding how data is spread out.

standard deviation

Standard deviation, represented by the symbol \( \sigma \), measures the dispersion of a set of data points around the mean. In essence, it tells us how spread out the values are in a data set.

• A low standard deviation indicates that the data points are close to the mean.
• A high standard deviation indicates that the data points are spread out over a large range of values.

For the quiz scores given, the standard deviation is 3. This means most of the scores are within 3 points of the mean score, either above or below. The more spread out the data, the higher the standard deviation.

mean score

The mean score, also known as the average, is calculated by adding all the data points together and dividing by the number of data points. The mean is a measure of central tendency and provides a general idea of where most values lie in a data set.
In the given exercise, the mean score is 10. This means that, on average, the students scored 10 on the quiz. The mean serves as a benchmark for other scores to be compared against. For instance, a score of 8 is below the mean, while a score of 15 is above the mean.

standard normal distribution table

A standard normal distribution table, or Z-table, is used to find the percentage of values below (or above) a certain Z-score in a standard normal distribution. The table lists the areas (or probabilities) under the curve to the left of a given Z-score.
For example, in the quiz scores exercise:

• A Z-score of \( \approx -0.67 \) corresponds to an area of 0.2514 to the left (25.14%) and 0.7486 to the right (74.86%).
• A Z-score of \( \approx 0.67 \) corresponds to an area of 0.7486 to the left (74.86%) and 0.2514 to the right (25.14%).
• A Z-score of \( \approx 1.67 \) corresponds to an area of 0.9525 to the left (95.25%) and 0.0475 to the right (4.75%).

The Z-table is fundamental in transferring Z-scores into more comprehensible percentages.

percentage area

The percentage area in the context of a normal distribution represents the proportion of data points that fall to the left (or right) of a given Z-score. This percentage helps in understanding the relative position of a score within the entire distribution.
In the exercise:

• The Z-score of -0.67 indicates that 25.14% of the scores are below 8, and 74.86% are above it.
• The Z-score of 0.67 indicates that 74.86% of the scores are below 12, and 25.14% are above it.
• The Z-score of 1.67 indicates that 95.25% of the scores are below 15, and 4.75% are above it.

By determining the percentage area, you can see how a particular score compares to the rest of the data, which is invaluable for statistical analysis.