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Z-Test for Population Proportions

Z-Test for Population Proportions

Author: Ryan Backman

Calculate z-statistic of a population proportion.

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Source: Rasmussen Reports

Video Transcription

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Hi. This tutorial covers hypothesis testing for population proportions. All right. Let's start with the situation here.

So a new telephone survey of likely Minnesota voters finds Barack Obama with 51% support to 46% for Mitt Romney. This survey of 500 likely voters in Minnesota was conducted on October 21, 2012 by a public interest group called Rasmussen Reports. OK? You can go to their website at It's a public opinion organization. And they frequently do political polls and things like that.

So does this study provide significant evidence that Barack Obama is favored by the majority of the population of Minnesota likely voters? So what we want to do is see if we can determine if the proportion of the population that supports Barack Obama is greater than 0.5. So being a majority. All right.

So a hypothesis test is appropriate here. So a hypothesis test for population proportions follows generally the same steps as z-tests and t-tests for population means. So let's investigate the four-process for a hypothesis test for a population proportion. All right.

So we're going to start with formulating our null and alternative hypotheses. And then we're also going to choose this significance level. All right. So we'll start with our null and alternate hypotheses. OK. And we want to assume a value for the null hypothesis, and assume a value for the population proportions.

So remember, we indicate the population proportion as p. And we're going to assume that p equals 0.5. And what we want evidence for is the alternative hypothesis, which we want to be p greater than 0.5. OK? If we're able to have evidence for this alternative hypothesis, we have enough evidence to say that a majority of the population supports the president.

OK. We also need to choose a significant level. Let's do a significance level of 0.05. That means that the probability of a type I error is only about 5% there. OK.

Next thing we need to do is check the condition, step 2. OK? So the sample observations were obtained randomly. OK? It didn't specifically say in the article that they were obtained randomly, but I would probably take a guess that when they're calling people, they are using some sort of random digit dialing. OK? So I would say it's probably pretty safe to say that one is satisfied.

2, observations must be independent. If sampling without replacement, n must be less than or equal to 10% of the population. OK. So if we're looking here, remember, we're just calling people. So I would say that one observation would not have an effect on another observation.

So what one person's opinion is, probably isn't going to affect another person's opinion. And since we are sampling without replacement here, n must be less than or equal to 10% of the population. OK. We're only talking about 500 voters out of the population of Minnesota, which is several million. So I would say that is certainly satisfied. And the sampling-- and so that's that one's good.

And condition 3, the sampling distribution is approximately normal. Now for proportions, all you need to do is test n times p0. OK? This is your assumed population proportion. So we just need that to be greater than or equal to 10. And m times q0-- all q0 is 1 minus p0.

So really, all we need to test is-- so if we start with n times p0, that's going to be 500, since we're surveying 500 voters. Our assumed population proportion was 0.5. OK. That equals 250, which is much greater than 10.

And then we do n times q0. So it would be 500 times 1 minus 0.5, which is also 0.5. So n times q0 is 250, also greater than 10. So we have all three of our conditions being met here. All right.

Now, step 3, we need to calculate a test statistic and compare to a critical value or find a p-value. All right. So the test statistic we're going to use here is z. We're going to use a z-score. And our z-score formula for a proportion is going to be p hat, our sample proportion, minus p0, which is our assumed population proportion.

And we're going to divide by square root of p0 q0 over n. Now the whole denominator of the z-score fraction is the standard error or the standard deviation of the sample proportions. OK? So this right here is known as the standard error in this case. All right, so let's go ahead and substitute in the values we know.

So our sample proportion remember, was 0.51. That was from the survey. Our assumed population proportion is 0.5. We're then going to divide that by 0.5-- big square root of 0.5 times 1 minus 5 over 500. So I'm going to do this in my calculator.

I think we'll do the whole denominator first, the standard error. So I'm going to do the square root of 0.5 times 1 minus 0.5 divided by 500. OK. That's going to be my standard error. And then what I'm going to do is I'm going to take my numerator, 0.51 minus 0.5. And then I'm going to divide by my standard error.

So I'm going to use the last answer function there. OK. I'm going to hit Enter. And my z-score is about 0.447. OK. OK. So then what I can do is I can either calculate a critical value based on my level of alpha, or I can find my p-value, and then compare my p-value to my value of alpha. All right.

So my p-value now-- since I'm using a z-statistic, I can use a normal distribution. And I'm I go again, use the calculator. So I'm going to do my normal CDF calculation. And I'm going to type in my z-score, 0.447.

Then I'm going to do comma. And then I'm going to use a very large number to represent positive infinity, because I'm doing an upper tail test. I'm going to hit Enter. And my p-value is about 0.327. And then I need to compare that to my value of alpha, which is 0.05.

And I can see here that my p-value is much greater than my value of alpha. All right? So in this case, I do not have evidence to support my alternative hypothesis. So I need to fail to reject my null hypothesis. All right.

So I'm going to do that in step 4. So step 4, what I'm going to say is that since p-value is greater than alpha, fail to reject the null. OK? So we decide whether to reject or not the null hypothesis. And then we also need to draw a conclusion. So now what I'll say is that there is not enough evidence to conclude that a majority of Minnesota voters will support President Obama. OK?

So again, p-value was greater than alpha. So I need to fail to reject. Then there is not enough evidence to conclude that a majority of Minnesota voters will support President Obama. All right. So that is step 4.

That concludes the step-- the four-step process in the hypothesis test for a population proportion. And that also concludes the tutorial on hypothesis testing for population proportions. Thanks for watching.


Terms to Know
Hypothesis Test for Population Proportions

A hypothesis test where we compare to see if the sample proportion of "successes" differs significantly from a hypothesized value that we believe is the population proportion of "successes."

Z-Test for Population Proportions

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

Formulas to Know
z-statistic of Proportions

z space equals space fraction numerator p with hat on top space minus space p over denominator square root of begin display style bevelled fraction numerator p q over denominator n end fraction end style end root end fraction