Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help. Today we're going to learn how to perform proportion hypothesis testing on vehicles with front license plates. Here's our problem statement: Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, and then state the conclusion about the null hypothesis as well as the final conclusion that addresses the original claim. Among 2072 passenger cars in a particular region, 241 had only rear license plates among 307 commercial trucks. 45 had only rear license plates. A reasonable hypothesis is that commercial truck owners violate laws requiring front license plates at a higher rate than owners of passenger cars. Use a 10% significance level to test that hypothesis. A) Test the claim using a hypothesis test. B) Test the claim by constructing an appropriate confidence interval.
OK, so Part A wants us to conduct a hypothesis test, and the first part of Part A asks us for the null and alternative hypotheses. So here we're going to look at not selecting answer option D because the null hypothesis here is not a statement of equality. Of the three answer options that remain, we need to look at the alternative hypothesis. To do that, we go back and look at the claim that's being made. And the claim is that commercial truck owners violate laws requiring front license plates at a higher rate than the owners of the passenger cars. So trucks are going to be greater than cars, but which is 1and which is 2. Well, if you read the problem statement, the cars are mentioned first, so they're going to be the first proportion. And then the trucks are going to be the second proportion. So trucks are greater than cars. That means 2 is greater than 1, which means 1 is less than 2. And the alternative hypothesis that says 1 is less than 2 is going to be this one right here. Well done!
Now the next part of Part A wants us to identify the test statistic. And to do that, we need to whip out StatCrunch. So I'm going to pop out StatCrunch here. And we'll resize this window to give us a better view of what's going on. OK, here in StatCrunch, I want to go to Stat --> Proportion stats (because we're dealing with proportions) --> Two sample (because we have two samples) --> With summary (because we don't have any actual data).
Here in the options window, we need to put in some statistics for our samples. The first sample is the one that was mentioned first, which is the passenger cars. So the number of successes --- we're going to consider a success having only a rear license plate, so I take that number right there from the problem statement --- 241. And I'm going to put in the total number of observations, which is the total number of cars --- 2072. I do the same thing with the trucks. And now down here under Hypothesis test, I want to make sure that this matches what we have earlier for our null and alternative hypotheses. Notice the format is written differently, but that's OK; they're algebraic equivalents. If I just take an add p2 to each side, I get the same thing. It's listed right over here. So I want to make sure that symbol is the same as this symbol, and now it is. And so I'm just going to leave that zero alone because that makes these two algebraic equivalents.
Now I'm ready to go and get my test statistic. And here we see the test statistic right here, second to last value there in that results window table. That's good. I'm asked to round to two decimal places. Nice work!
Next we're asked to identify the P-value. The P-value is right next door to the test statistic; it's that last value there in that results window table. And I'm asked to round to three decimal places. Good job!
Now the fourth part of Part A says, "State the conclusion about the null hypothesis as well as the final conclusion that addresses the original claim." Well, if I go back and compare my P-value with my significance level --- and let's see, where do we have our significance level? I'm looking, I'm looking, I'm looking. Wow, I don't see where it --- oh, it's right here. Duh! Right in front of you --- 10% significance level. So if I come down here, I look at 10% significance level. My P-value is 6%, so we're under the significance level, which means we're inside the region of rejection. Therefore, we're going to reject the null hypothesis. And whenever you reject the null hypothesis, there's always sufficient evidence. Excellent!
Now Part B asks for a confidence interval from the same data. So I could go back through all those motions again, but I'm lazy. So I'm going to come back up here, click on Options --> Edit, and then down here I'm gonna switch this radio button to Confidence interval. And they don't specify a confidence level, so we have to determine the appropriate one. If we've got a 10% significance level, that would mean alpha is 10%, but we've got two samples. So I've got to subtract two alpha, and that's going to give me an 80% confidence level.
Now I got my upper and lower limits for my confidence interval, and I can place those in here. We're asked to round to four decimal places. It's making me count today! There's my lower limit. Now I get to put in the upper limit. Fantastic!
And now the second part of Part B asks us to interpret the confidence interval, which we can see here. We look at their confidence interval. It does not contain zero, and so, because it does not contain zero, that means one of these proportions is always going to be bigger than the other. Since the entirety of the confidence interval is in the negative region of our number line, this difference is always going to be negative. So that means p2 is always going to be greater than p1. And that was the actual claim that we were making, because 2 corresponds with the trucks, 1 corresponds with the cars. And so, 2 being greater than 1 means that the trucks are going to be greater than the cars, which means their rate of noncompliance is higher than the rate of the owners of the passenger cars.
So because the confidence interval does not contain zero, there is a significant difference between the two proportions. Because there's a significant difference, that means we can reject the null hypothesis, because the null hypothesis is a statement of equality. And every time we reject the null hypothesis, there is sufficient evidence. Excellent!
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Frustrated with a particular MyStatLab/MyMathLab homework problem? No worries! I'm Professor Curtis, and I'm here to help.