Hypothesis Testing Large Independent Samples

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• 0:01 Hypothesis Testing
• 0:47 Large Independent Samples
• 1:53 Z-Test
• 3:10 Example
• 4:57 Lesson Summary

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Lesson Transcript
Instructor: Artem Cheprasov
In this lesson, you're going to learn about hypothesis testing of large independent samples. You'll learn about the assumptions in such a hypothesis test, and we'll work through an example together.

Hypothesis Testing

Let's say that you're in a supermarket browsing for bottles of orange juice. Your favorite orange juice manufacturer claims to have 100 ounces of orange juice in each bottle. If we were to take a sample of 100 bottles of this orange juice and find that the mean amount of orange juice in these bottles is 99.93 ounces, does that mean that on average there is less than 100 ounces in every bottle of this orange juice, and that the company is lying to the nation about the amount of orange juice it has in each bottle? Maybe. Maybe not.

This is what hypothesis testing sets out to discover. There are many different aspects to hypothesis testing, and this lesson focuses on hypothesis testing of large, independent samples.

Large Independent Samples

Let's say we are comparing the means µ1 and µ2 of two different populations and that we want to compare the difference of these two population means using something called the z-test, which we'll get to shortly. To do this, the following assumptions must be true.

One: the samples have to be independent of one another, meaning the sample from one population isn't related to the sample from the other population. Two: the samples must be obtained from normally distributed populations with known standard deviations; or the sample sizes have to be greater than or equal to 30.

Standard deviation is the amount of variation in a set of data values. A large independent sample is a sample size that is greater than or equal to 30. In such a scenario, we hold the following to be true: the null hypothesis, H0, tells us that µ1 is equal to µ2. The alternative hypothesis, HA, tells us that µ1 is not equal to µ2.

Z-Test

Assuming both of our randomly selected samples have a size of at least 30 and that they are independent of one another, a z-test can be used to determine any difference between two population means. We don't need to know the actual population means, because we are testing the difference between the population means based on the means of the samples. The equation for the z-test is as follows:

Where (x1 - x2) represents the observed difference in our respective sample means, where (µ1 - µ2) is the expected difference in populations means, which is 0, since H-naught tells us that µ1 is equal to µ2, which alternative means µ1 - µ2 = 0.

It also tells us where the denominator represents the standard error. In the denominator the symbol Sigma represents the standard deviation, and the letter n stands for each respective sample size. The standard deviation, in this case, represents the variability of individual observations around their mean, while the standard error, in simple terms, represents the variability of the sample mean across different samples.

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