Question

A test is normally distributed with a mean of 70 and a standard deviation of 8 . (a) What score would be needed to be in the 85 th percentile? (b) What score would be needed to be in the 22 nd percentile?

Short answer

85th percentile: ~78.29, 22nd percentile: ~63.82

Step-by-step solution

Understand the Problem

To find the score needed for a certain percentile in a normally distributed test, we need to identify the 85th and 22nd percentiles based on the mean and standard deviation provided.

Use Z-Score Formula

We use the formula for the Z-score: \[ Z = \frac{X - \mu}{\sigma} \]where \(X\) is the score, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

Identify Z-Score for the 85th Percentile

From a standard normal distribution table, the Z-score corresponding to the 85th percentile is approximately 1.036.

Calculate Score for 85th Percentile

Use the Z-score for the 85th percentile in the Z-score formula: \[ 1.036 = \frac{X - 70}{8} \]Solve for \(X\):\[ X - 70 = 1.036 \times 8 \]\[ X = 70 + 8.288 \]\[ X \approx 78.29 \]

Identify Z-Score for the 22nd Percentile

From a standard normal distribution table, the Z-score corresponding to the 22nd percentile is approximately -0.772.

Calculate Score for 22nd Percentile

Use the Z-score for the 22nd percentile in the Z-score formula: \[ -0.772 = \frac{X - 70}{8} \]Solve for \(X\):\[ X - 70 = -0.772 \times 8 \]\[ X = 70 - 6.176 \]\[ X \approx 63.82 \]

Key concepts

Normal Distribution

The normal distribution is a fundamental concept in statistics which is often referred to as the bell curve due to its distinct shape. It represents how the values of a variable are distributed. In a perfectly normal distribution:

• The mean, median, and mode of the data are equal.
• The curve is symmetrical around the mean.
• About 68% of the data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.

This type of distribution is commonly seen in various natural and social phenomena, such as test scores, heights, and measurement errors. Due to its wide applicability, understanding normal distribution is crucial in many fields of study.

Z-score

A Z-score is a statistical measurement that describes a value's position relative to the mean of a group of values. By using Z-scores, you can determine how far away a data point is from the average, expressed in terms of standard deviations.

• Positive Z-scores represent values above the mean.
• Negative Z-scores indicate values below the mean.

The formula for calculating the Z-score is:\[ Z = \frac{X - \mu}{\sigma} \]where:

• \(X\) = the value being analyzed,
• \(\mu\) = the mean of the dataset,
• \(\sigma\) = the standard deviation.

This measure allows comparison between different data points from different datasets, providing a clear understanding of where a value stands in relation to the typical range.

Mean and Standard Deviation

The mean and standard deviation are key statistical concepts frequently used in data analysis.
- The **mean** is the average of a set of numbers, calculated by summing all the values and then dividing by the count. It provides a central value of the dataset and gives a quick insight into the general tendency of the data distribution.
- The **standard deviation** measures the amount of variation or dispersion in a set of values. A low standard deviation means the values tend to be close to the mean, whereas a high standard deviation indicates the values are spread out over a wider range. In a normally distributed dataset:

• Most values will lie close to the mean.
• The standard deviation determines the shape and width of the distribution.

Together, they form the backbone of understanding data distribution and variability, essential in interpreting various statistical measures.

Percentile Calculation

Percentiles are a way of understanding the relative standing of a particular value in a data set. They indicate the percentage of data points that fall below a certain value.
For example, being in the 85th percentile means that 85% of the values in the dataset are below your score.

• To find a specific percentile in a normally distributed dataset, you use the Z-score.
• The Z-score corresponding to a desired percentile can be found in a standard normal distribution table.

Once you have the Z-score, you can calculate the score using:\[ X = \mu + Z \times \sigma \]where:

• \(X\) is the score.
• \(\mu\) is the mean.
• \(\sigma\) is the standard deviation.

This calculation allows you to translate percentile rankings into actual scores, making it easier to understand where you stand within a distribution, like seeing what test score corresponds to a particular percentile.