Performing a Sign Test In Exercises 7–22, (a) identify the cla...

Question

Performing a Sign Test In Exercises 7–22, (a) identify the claim and state Ho and Ha, (b) find the critical value, (c) find the test statistic, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim.

Unit Size A renters’ organization claims that the median number of rooms in renter-occupied units is four. You randomly select 120 renter-occupied units and obtain the results shown below. At , can you reject the organization’s claim? (Adapted from U.S. Census Bureau)

[IMAGE]

Accepted Answer

All right, hi everyone. So this question says, a transport agency claims that the median one-way commute time for employees is 30 minutes. You randomly select 60 employees and record whether each commute time is less than, equal to or greater than 30 minutes. Part one says identify the claim and stateh not and H sub a. And part two says, at the 0.05 significance level, can you reject the agency's claim? Interpret the decision. And here we have 4 different answer choices labeled A through D. So, here, let me go ahead and scroll down past the answer choices to open up some space. Then Let's begin with the claim and the hypothesis. Or hypotheses According to the text of the question, the claim. Is that the population median commute time represented here as tau is equal to 30 minutes. Now, notice the wording of the question. The question is asking us for part two, if we can reject the claim that is being made. Because we're testing if we can reject the claim, the claim in this case corresponds to the null hypothesis. So each knot Is that tau is equal to 30. Which means that H A is the opposite. So it would be that tau is not equal to 30. From here, we can test the claim at alpha equals 0.05. But before we do that, let me just go ahead and scroll. So we can see The results here of the survey. Based on this data, we can see that there are 3 employees who have commute times exactly equal to 30 minutes. These are the ties, and these have to be removed from the data set. So, the effective sample size that we'll call N is equal to the original 60 subtracted by the 3 ties, which gives you 57. And now we counted the sights. So let's say that S positive represents the number of commute times that are greater than 30 minutes. S, therefore, is equal to 35. And so with this information, let me go ahead and scroll down to where I have more space. Recall that we're using the normal approximation here. So under H knots positive. Follows a binomial distribution. Where N is equal to 57. And P is equal to 0.5. So here we can approximate with capital Z. Equals S positive subtracted by N divided by 2. And all this is divided by the square root of N divided by 4. So, based on the information that we have, Z is equal to 35 subtracted by. 28.5. And divided by the square root of 57 divided by 4. And this equals approximately 1.72. Now, we can find the critical values for a two-tailed test at alpha equals 0.05. Those critical Z values. Are equal to positive and negative 1.96. And now we're left to make a decision. Now, our calculated value for Z. Or specifically the absolute value of that number, that's 1.72, is less than 1.96. And because it's less than 1.96 or critical value, ultimately we fail. To reject The null hypothesis. So what does that mean? How do you interpret this decision? Well, what this means is that at the 5% level of significance, there is not enough evidence to conclude that the true median commute time is different from 30 minutes. So now I can scroll back up to view the answer choices. And based on the information that we just obtained, the correct answer is option C. And so with that being said, thank you so very much for watching and I hope you found this helpful.

Key Concepts

Hypothesis Testing

Sign Test

Critical Value

Watch next