A city government claims that no more than

Question

25 percent sign

of households have solar panels. A researcher suspects the rate is actually higher and surveys

200

households, finding that

62

have solar panels. Test if there is evidence that more than

25 percent sign

of households have solar panels using

alpha equals 0.01

.

$H sub 0 colon$ _________ $H_{a} :$ _________

$x =$ ________ $n =$ ________ $P$ -value: ______

Because $P$ -value [ < | > ] $α$ , we [ REJECT | FAIL TO REJECT ] $H sub 0$ , there is [ ENOUGH | NOT ENOUGH ] evidence to suggest…

Accepted Answer

Welcome to The Practice. In this problem, it says a city government claims no more than 25% of households have solar panels. A researcher suspects the rate is actually higher and surveys 200 households, finding that 62 have solar panels. We'd like to test if there's evidence that more than 25% of households have solar panels using alpha equals 0.01. So let's begin by writing our hypotheses. The null hypothesis is going to be that the true population proportion of households with solar panels is what we expect it to be, and we can see that that expectation or benchmark is 25%. So our null hypothesis is that p is equal to 0.25 or 25%. The alternative hypothesis is what we're trying to test or find evidence for, and we're trying to see if actually maybe more than 25% of households have solar panels. So we wanna see if maybe p is greater than 0.25 or 25%. Now we are gonna pull out our TI 80 fours in a second, but it is a good idea to get some key values from the problem that we're going to need to plug in. We will need x, which is going to be the number of households with solar panels in the sample, which we can see is 62, and n, the total sample size, which is 200. Now that we have those values, we are ready to pull out our calculator, so let's get those out. For step one, we're going to click on the stat button, then scroll two to the right to the test menu, and we wanna go down to option five, which is the one dash prop z test function and click enter. And that is step one all set. For step two, we just wanna enter in some key information from the problem, starting with p sub zero. As a reminder, p sub zero is the expected population proportion, which we can get from the null hypothesis as 0.25. So let's put in 0.25 for p sub zero. Next, we're gonna put in x, which we found to be 62, and n, which we found to be 200. Next, we need to choose the correct alternative hypothesis. We can see that our alternative hypothesis is that p is greater than 0.25. As a reminder, 0.25 is p sub zero, so we wanna choose greater than p sub zero as our alternative hypothesis. Once we've selected it, we can scroll down to calculate and click enter, and that is step two all set, and the results of the test will appear in the viewing window. We can see that the p value is approximately 0.03, and we want to compare it to our alpha of 0.01. Now, 0.03 is greater than 0.01, so our p value is greater than alpha, meaning that we fail to reject the null hypothesis, so there is not enough evidence to suggest that p is greater than 0.25, our alternative hypothesis. In the context of the problem, we can say that there is not enough evidence to suggest that the true proportion of households with solar panels is greater than 0.25 or 25%. Great work.

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