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Z-Test for Population Means

Z-Test for Population Means

Author: Katherine Williams

Calculate z-statistic of a population mean.

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This tutorial talks about the z-test for population means. The z-test for population means is one that we don't use very often. It's unlikely that we would know the population standard deviation without knowing the population mean, and when we do, there's a simpler way of doing hypothesis testing.

Now we often use technology when doing the z-test for hypothesis-- sorry, for population means, and a standard error for the z-test of a population mean is the standard deviation divided by the square root of the sample size. Now we have a series of steps when we are finding the z-test for population means that we follow.

Here are the steps. First, we need to formulate the null and alternative hypotheses and choose a significance level. Then we need to check that the conditions of the hypothesis are met for the random sample we use. Thirdly, we need to calculate a test statistic and compare it to a critical value, or find the p-value and compare that to the significance level, and then we need to decide whether or not to reject-- sorry, whether to reject or not reject the null hypothesis and then draw a conclusion.

Now here we are checking the conditions of the hypothesis test, these are the same conditions we've seen over and over again-- was the data randomly collected, are the observations independent, and is the sampling distribution approximately normal? Here, we're calculating a test statistic. In this case we're doing a z-test, so this is the test statistic that we would use-- z equals the sample mean minus the population mean divided by the standard deviation, which is divided by the square root of the sample size. Let's look at an example.

In this example, there's a school of 400 students. The spelling quiz scores for 50 students were randomly selected for analysis. The sample mean was 7.5 questions correct with a standard deviation of 0.5, and we want to test the hypothesis that the school performed worse than the state average of 7.7 questions correct.

So first, our null hypothesis is that the mean is, in fact, 7.7. The alternative hypothesis is that the mean is less than 7.7. The significance level is 0.05. Next, we need to check that the conditions are met, and yes, the quizzes were selected randomly, and the selections are done independently, and the other conditions are met as well.

Next, we need to calculate a test statistic. And since we're doing a z-test, we're going to do a z-statistic. So we have 7.5 minus 7.7 divided by the standard deviation for the sample is 0.5, and then divide that by the square root of our sample size of 50. When we calculate that-- put that into our calculator, we get negative 2.83. Now using a z-table, we can find the p-value, and we find 0.0023. Now that 0.0023 is less than our significance level of 0.05, so we're able to reject the null hypothesis that these school mean is 7.7.

So this has been your tutorial on z-tests for population means.

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Terms to Know
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
Z-Statistic for Population Means

z space equals space fraction numerator x with bar on top space minus space mu over denominator begin display style bevelled fraction numerator sigma over denominator square root of n end fraction end style end fraction