GMAT

Sub Topics

Understanding Bar Graphs And Pie Charts Summarizing Categorical Data Using Tables How to Calculate Percent Increase With Relative Cumulative Frequency Tables What is a Two Way Table Make Estimates And Predictions From Categorical Data What is Quantitative Data Definition Examples What is a Histogram in Math Definition Examples What Are Center Shape And Spread How to Calculate Mean Median Mode Range Describing The Relationship Between Two Quantitative Variables Reading And Interpreting Line Graphs Making Estimates And Predictions Using Quantitative Data z Test Formula Example Binomial Distribution Definition Formula Examples Range of Values Definition Overview Data Abstraction Definition Example Types of Statistical Analysis How to Draw a Trend Line How to Find The Equation of a Trend Line

Main Topics

Lessons and MCQs

Select Test Select Question Types

GMAT: Statistical Analysis With Categorical Data
z Test Formula Example

In statistics, z tests are calculations that can be used to compare sample and population means to see if there is a significant difference. In this lesson, you will learn about these useful statistical methods.

z Test Formula Example

Z tests and Drive Thrus

Fast food chain Speedy Burger claims their customers can expect to complete a drive-thru order, from the time they place the order until they receive their food, in 90 seconds with a standard deviation of 5 seconds. This means that the average customer can expect to get a delicious meal in about a minute and a half. Chill Hot Dog, a competing fast-food company, wants to disprove this claim, so they send some spies with stop-watches to time a random sample of 50 customers ordering through Speedy Burger's drive-thru. Through the observations, they find that the mean time to receive a completed order is 102 seconds. Is this enough evidence to contradict Speedy Burger's claim? To find out, a z-test can be conducted.

Z tests are very useful calculations when comparing sample and population means to determine if there is a statistically significant difference. After all, the difference between the sample and population means may be due to random chance. These calculations can be used to prove or disprove claims or hypotheses. They require a random sample from a population that is normally distributed and has a known mean (average). Z-tests are most useful in large populations, where the sample size is greater than 30. Let's learn more about these useful statistical methods and the formulas to use.

Different Types

There are a variety of z-tests that can be used for different purposes, but two of the most common are the one-sample z-test and the two-sample z-test. As their names suggest, the one-sample z-test compares the mean for one sample and the two-sample z-test compares two. Z-tests are used mainly when there are larger samples (n>30), but no matter which z-test is used, they all require data with a normal distribution, which means that the data is distributed evenly around the mean.

One-Sample Z-Test Formula

Let's explore an instance where a z-test could be used to try to disprove a claim. Remember Speedy Burger's drive-thru timing claim? They have decided to prove to Chill Hot Dog, once and for all, that no matter what their spies reported back, Speedy Burger's 90 second claim is true. To do so, they perform a few z-test calculations.

Since they will be comparing their existing data for the drive-thru timing with that of the random sample of customers that their competitors observed, they should use the one-sample z-test. The formula is:

Where:

kkk

Substituting the data into the formula yields a z-score, called a critical value. The z-score is the value we look at to determine whether the hypothesis is correct. The z-score here is found to be 16.97. What does this z-score mean? It means that the particular sample is 16.97 standard deviations above the mean, meaning that for these particular samples, on average, have values that are above the standard mean. For Speedy Burger, it means that, on average, they don't meet the 90 second claim. If the z-score was low enough, even if it was above the standard mean, it would still mean that Speedy Burger does meet the 90 second claim. But, unfortunately, Speedy Burger's z-score is much higher.

Two Sample Z Test

When there is a need to compare two normally distributed but independent populations, and the standard deviations are known, we can use the two-sample z-test. Say, for example, we want to determine who is more likely to vote in an election: men or women. This would involve a comparison of two independent populations, so the two-sample z-test would be appropriate. The formula for the two-sample z-test is similar to the one for just the one sample:

The delta symbol here represents the hypothesized difference between the two samples. Say we have this information for the men and women voting:

If the hypothesis is that men and women are equally likely to vote, then the delta here is zero, since the hypothesis is that there is no difference. Plugging in these values using Sample 1 as men and Sample 2 as women, we get this z-score:

kkk

We get a negative z-score since we're subtracting the women from the men and the sample mean for the women is higher than the men. As far as understanding the z-score, it doesn't matter whether it is negative or positive, since our z-score is lower than the standard mean and it's not a very small number, it's telling us that our hypothesis that men and women both vote equally is not true.

In reality, however, the two-sample z-test is not commonly used because two population standard deviations are usually not known.