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Common Core: S.IC.5

# Calculating Z- and T-Test Statistics

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##### Description:

This lesson will explain how to calculate both z- and t-test statistics.

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Tutorial

## Video Transcription

In this tutorial, we're going to practice calculating Z- and T-test statistics as well as identifying when to use what formula based off the information that's been given to you. So the first formula that we're going to look at involves quantitative data and knowing the population standard deviation. When we know the population standard deviation, we can use the normal distribution. And that's why it's a Z-score.

When we don't know the population standard deviation, though, for quantitative data, we're going to have to approximate with a student's T-distribution. And therefore, we're going to be finding a T-score based off of this formula.

But what about qualitative data? When it's qualitative data and we know the population standard deviation, we're going to use a normal distribution. So we get to use a Z-score. But we're going to be dealing with proportions instead of mean. So we're going to use this following formula. I'm going to go through an example of each. And we're going to, again, look at how to identify which of these three formulas to apply.

Let's look at our first example. Approximately 10% of the population is left-handed, with a standard deviation of 3.13%. 100 people were randomly selected and 14 claimed to be left-handed. So we need to find the Z-test statistic for this data set.

This type of data is qualitative data. You're answering either yes or no. You are either left-handed or you're not left-handed. And you're placing your answers into categories, which is why it's also called categorical data.

We also know the population standard deviation at 3.13%. Therefore, we can use this formula to calculate our Z-test statistic. So let's go ahead and find out the values necessary to calculate my Z-score.

My p hat-- that is the proportion of successes from our sample. And in this case, a success is being left-handed. So that is 14 out of 100, which is 14%. Now, p sub 0 is the population number proportion of successes, which is 10%. So 0.10. And q would be the complement to that. So people who are right-handed, which would be 90%. And our sample size was 100.

So let's go ahead and plug in these values. So we've got our 14% minus the population proportion of 10%. We're going to calculate underneath the standard error. So 0.1 times 0.9 all over my sample size. And take the square root. And we end up getting a Z-test statistic of 1.33.

So if we were to put this on a normal distribution-- and we can use a normal distribution because we know the population standard deviation. And this distribution is centered at 10%. Our sample rendered 14% of people being left-handed, which was 1.3 standard deviations above the mean.

For our next example, we're looking at the average weight of newborn babies, which we know is 7.2 pounds. A local hospital has recorded the following weights of randomly selected newborn babies born during a certain week. And we need to calculate the T-test statistic for this data.

Well, this is quantitative data. Because we're looking at pounds. We also do not know the population standard deviation. Therefore, we have to use a student's T-distribution to approximate the normal distribution. So we have to use the following formula to calculate our T-test statistic.

So let's go ahead and find all of these values. So I'm going to show you how to do this in your calculator. And then I will also show you how to do it in Excel. So first we're going to enter all of this data into a list in our calculator. So hit the Stat button. We're going to hit Enter. And under List 1, we're going to insert all of these weights. So I have 7.2 pounds, 6.4, 8.1, 7.7 pounds, 6.9 pounds, 6.8 pounds, 9.2 pounds-- that's a big baby-- and 8.3 pounds. Go ahead and exit out of this screen by hitting Second, Mode.

And now we're going to get the summary statistics based off of this data. So again, hit Stat. Scroll over to Calc. And we're interested in this first function, which is one variant statistics. And we're going to look for that under List 1. So hit Second, 1. You can see that L1 in orange in the upper left-hand corner. Hit Enter. And we get all of these summary statistics based off that data set.

So my average weight for the babies based off of this sample is 7.575. Let's see. They are saying in the problem from you that the average weight of newborn babies is 7.2 pounds. And remember that population statistics are denoted with Greek letters, whereas our sample statistics are just noted with x and s.

So the s of x, the standard deviation of our sample, is 0.9285. Remember, we're not going to use sigma of x. That's the population standard deviation. And ours is based off of a sample. And we had a sample size of 8 babies.

So let's go ahead and plug these values into our formula. So I have 7.575 minus the average of 7.2 pounds all over 0.9285 divided by the square root of 8. And that gets me a T-test statistic of 1.142.

So if I were to put this on my student's T-distribution, that would be centered at 7.2 pounds. Our sample had an average weight of above 7.2 pounds. So it should be a positive T-score. And it should fall in the upper right-hand part of our distribution. So it's about right here that corresponds to our sample.

In our final example, we're still looking at the average weight of newborn babies. But in this case, we know the population standard deviation. So they're telling us the average weight is 7.2 pounds, with a standard deviation of 1.1 pounds. At a local hospital, they've recorded the weights of all 285 babies born in a month. And the average weight was 6.9 pounds. So we need to find the Z-test statistic for this data set.

Now, because we know the population standard deviation and its quantitative data, we can use the normal distribution and find a Z-score. So we're going to use this formula to calculate our Z-test statistic. So the average weight from our sample-- that's denoted with an x bar-- was 6.9 pounds. Let's see. Mu is the population average, which was 7.2 pounds. And sigma is the population standard deviation-- was 1.1 pounds. And we had a sample size of 285.

So let's go ahead and calculate the Z-score. So I have 6.9, which is our sample mean, minus the population mean of 7.2 divided by the population standard deviation divided by the square root of our sample size. And this gives us a Z-score of negative 4.604. And we should expect to get a negative Z-score. Because our sample was less than the population mean.

So if I were to put this on a normal distribution, it's centered at the population mean, which is 7.2 pounds. The average weight of the babies at the hospital was less than 7.2. So I should fall in the lower part of my distribution. And we fell really far below. We're all the way down here somewhere at the negative 4.604. So this hospital-- the average weight of their babies was definitely far below the average weights of the babies of the population. I hope this tutorial was helpful in practicing calculating Z- and T-tests.

## Additional Practice Problem

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Terms to Know
t-test for population means

A hypothesis test for a mean where the population standard deviation is unknown. Due to the increased variability in using the sample standard deviation instead of the population standard deviation, the t-distribution is used in place of the z-distribution.

z-test for Population Proportions

A type of hypothesis test used to test an assumed population proportion.

z-test for population means

A hypothesis test that compares a hypothesized mean from the null hypothesis to a sample mean, when the population standard deviation is known.

Formulas to Know
t-statistic

z-statistic of Means

z-statistic of Proportions