Perception and Reality In a presidential election, 308 out of ...

Question

Perception and Reality In a presidential election, 308 out of 611 voters surveyed said that they voted for the candidate who won (based on data from ICR Survey Research Group). Use a 0.05 significance level to test the claim that among all voters, the percentage who believe that they voted for the winning candidate is equal to 43%, which is the actual percentage of votes for the winning candidate. What does the result suggest about voter perceptions?

Accepted Answer

Welcome back, everyone. A survey of 750 college students found that 405 believe they are above average in academic performance. The actual percentage of students with above average grades is 50%. At the 0.05 significance level, test the claim. That the proportion of students who believe they are above average matches the actual percentage. What does this indicate about student perceptions? No, If we're going to test this claim, OK, then we would need to first set up the hypothesis. We're testing if the proportion of students who believe they are above average matches the actual percentage. So our non hypothesis is that they are not, that is the proportion is equal to 0.5 or average. Which means then that our alternative hypothesis is that the proportion is not equal to 0.5. So, now, if we're going to test this, we'll need to use the test for proportions, OK? Now, what do we know that can help us? Well, so far we know that the sample size N is 750. The number who believe they are above average, OK, is 405. So let's call that X. Which means then that the sample proportion P will be equal to 4005 divided by 750 or 0.54. The population proportion under the null hypothesis, PO is going to be equal to 0.5, OK? And we're told that we're testing at 0.05 significance level. So, alpha equals 0.05. Now, if we're going to use the if we're going to calculate the test statistic for proportions, we can use the formula for a Z test for proportions, OK? And remember that Z is going to be equal to our sample proportion minus our population proportion divided by the standard error. Now we know that the standard error is equal to the square root of the population proportion multiplied by 1 minus the population proportion divided by n. And if we calculate that, then it's gonna be the square root of 0.5 multiplied by 1 minus 0.5 divided by 750. When we go ahead, then this should be equal to 0.01826, which means that our Z value, our test statistic, is gonna be 0.54 minus 0.5 divided by 0.01826. And that is equal to 2.19. Now we have the test statistic, we need to compare it to the critical Z value for this two-tail test at the 0.05 significance level, OK? So for a two-tailed test. OK. Add alpha equal to 0.05. If we split the significance level, then the critical Z value, OK. Is going to be equal to plus or minus 1.96 for this two-tailed test. Now notice here. That our test statistic 2.19 is greater. Let me write that here. Our test statistic Z. Is greater than our critical Z value, OK. Critical Z value of 1.96. So in that case, that means the test statistic falls into the rejection region and we can reject the null hypothesis. So what does that mean? Well, let's write our statement here, our conclusion that we can make. No, I, I prefer to type it's a little easier to type. No, this tells us then that there is sufficient evidence. At the 0.05 significance level. To support the claim, OK. That the proportion of students. Who believe they are above average is different. From the actual 50%, OK? And what does this mean then? Remember, or the rest of our question ask us what does this indicate about student perceptions? Well, this suggests that student perceptions, it's also right out here, this suggests student perceptions do not match reality, OK? Because more students believe they are above average than is actually the case. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Hypothesis Testing

Significance Level

P-Value

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