Question

Because \(P(z<.44)=.67,67 \%\) of all \(z\) values are less than \(.44\), and \(.44\) is the 67 th percentile of the standard normal distribution. Determine the value of each of the following percentiles for the standard normal distribution (Hint: If the cumulative area that you must look for does not appear in the \(z\) table, use the closest entry): a. The 91 st percentile (Hint: Look for area \(.9100 .\) ) b. The 77 th percentile c. The 50 th percentile d. The 9 th percentile e. What is the relationship between the 70 th \(z\) percentile and the 30 th \(z\) percentile?

Short answer

The 91st percentile corresponds to a z-score of around 1.34. The 77th percentile corresponds to a z-score of around 0.75. The 50th percentile corresponds to a z-score of 0. The 9th percentile corresponds to a z-score of around -1.34. The 70th percentile is the negative of the 30th percentile in a standard normal distribution because percentile values mirror across the mean.

Step-by-step solution

Understand what is being asked

The overall goal is to find the z-scores that correspond to the 91st, 77th, 50th, and 9th percentiles. Additionally, the relationship between the 70th and 30th percentile should be determined.

Locate z-score in table for each percentile

Using a z-table, look up the z-values based on the cumulative area. For the 91st percentile, since we need an area of 0.9100, locate this in the body of the z-table. Repeat this process for the 77th and 9th percentiles.

Determine 50th percentile

The 50th percentile in a standard normal distribution is always 0. This is because half of the values are below the mean and half are above in a standard normal distribution.

Determine relationship between 70th percentile and 30th percentile

The relationship between the 70th and 30th percentiles can be found by realizing that in a standard normal distribution, percentile values are mirrored around the mean (50 percentile). If \(\) is the 30th percentile, then \(-\) is the 70th percentile.

Key concepts

Standard Normal Distribution

The standard normal distribution is a special type of bell-curved distribution that is essential in statistics. It is characterized by a mean of 0 and a standard deviation of 1. This means that most of the data values are spread around the mean value, with fewer values found as we move away from the mean in both directions along the x-axis.

This distribution is used as a reference to understand how data compares to a standard or 'normal' case. For instance, if a student's score in a test is above the mean, it might suggest they performed above average compared to their peers. Furthermore, the area under the curve corresponds to probability and helps us to determine the likelihood of a particular outcome occurring.

Practically all the values (around 99.7%) in a standard normal distribution lie within three standard deviations from the mean. This property is known as the empirical rule or the 68-95-99.7 rule, and it is useful for assessing probabilities in a variety of contexts.

Z-scores

A z-score is a numerical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean. Z-scores are crucial for understanding where a particular score stands in comparison to a standard normal distribution.

More formally, it is calculated with the formula: \[ z = \frac{(X - \mu)}{\sigma} \] where \(X\) is the value in question, \(\mu\) is the mean of the dataset, and \(\sigma\) is the standard deviation. Z-scores are particularly helpful because they allow us to compare scores from different distributions by standardizing them, hence providing a basis for making statistical inferences.

Z-table

The z-table is an invaluable tool for statisticians. It contains the cumulative probabilities of the standard normal distribution up to a given z-score. This table is commonly used to find the percentage of values that lie below a certain z-score.

When you look at a z-table, you're seeing the probability that a standard normal variable is less than or equal to \( z \). For finding percentiles, which show the relative standing of a value within a distribution, the z-table provides a straightforward way to translate a percentile rank into a z-score. For example, if you're looking for the 91st percentile, you would seek the z-score associated with 0.9100 cumulative probability in the table.

Cumulative Distribution Function

The cumulative distribution function (CDF) is a function that yields the probability that a random variable is less than or equal to a certain value. In the context of a standard normal distribution, the CDF gives us the area to the left of a z-score, and this area represents the probability that the variable takes on a value less than or equal to that z-score.

For our exercise, the CDF equates to the cumulative probability associated with each percentile. For instance, the 50th percentile represents the median, which, for a standard normal distribution, has a CDF of 0.5; this corresponds to a z-score of 0. Understanding the CDF is essential for interpreting percentiles in statistics, as it tells us what proportion of the data falls below a particular threshold.