Performing a Sign Test In Exercises 7–22, (d) decide whether t...

Question

Performing a Sign Test In Exercises 7–22, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim.

Social Media A research group claims that the median age of the users of a social media website is greater than 30 years old. In a random sample of 24 users, 11 are less than 30 years old, 10 are more than 30 years old, and 3 are 30 years old. At , can you support the research group’s claim? (Adapted from Pew Research Center)

Accepted Answer

Hello and welcome back, everyone. The next question says, a volunteer group claims that the median number of weekly donations they receive is greater than 25. A random sample of 16 weeks gifts, 4 weeks had donations above 25. 11 weeks 11 weeks had donations below 25, 1 week had exactly 25. Use a right-tailed sign test to test the claim at alpha equals 0.05. So we know we can look at our sample here, we're going to eliminate the ties, so that one week that had exactly 25. And rather than say a sample of 16 weeks, we'll add up a number of weeks that had donations above or below 25. And that is 15 weeks, obviously. So our sample N is equal to 15. And let's think about our hypotheses here. So, our no hypothesis in this case, we're talking about median, and when we're comparing median, we look at The null hypothesis being that the median is exactly at that point, so the null hypothesis would say that the median. Is equal to 25. While the alternative hypothesis put forward would be that the median is greater than 25. So we know we need a right tailed test in this case, which we're told anyway. And so we have our end value. We need our P value, but we're talking about medians. So we know our positive and negative result, our positive result would be above 25. And our negative result would be below 25. So our probability of a positive result. It is going to equal 0.5. It's a median, so half the results are going to be above that and half below it. So a result has an equal chance of being above or below. We have a right-tailed scientist, so we're going to look for the number of positive signs, weeks above 25. So that would make our test statistic. And we know we were told that we had 4 weeks, but it's a donation above 25, so our test statistic X equals 4. So, we have our alpha given to us. Alpha is 0.05, and let's look at our So a way that we test, so we're looking for the smallest number. The smallest number of Xs of where the positive result is significantly large, if the null hypothesis is true. So, we're going to use the binomial distribution. We've got a small number of samples, and we have 2 possible results, and we know for that we need our values of N, which we said was 15 and P. Which is equal to 0.5. Oops. But for a critical value we're going to look. For P we'll look at alpha equaling 0.05. So Using either a calculator or binomial table, we would say that the critical value in this case. Is 12. So we have our critical value of 12. Our test statistic of 4. And we can see that our test statistic is less than our critical value, 4 is less than 12, and therefore we cannot reject the null hypothesis. So again, not sufficient evidence to reject the null hypothesis. Meaning there is insufficient evidence for the claim, they say. Insufficient evidence. For that claim. So, once again, on our problem, we had a claim of a volunteer group that the median number of weekly donations is greater than 25. We had a sample with donations above, below, and at 25, we eliminated our zeros. And then we were told to use a right-tailed scientist to test the claim in alpha equals 0.05. So we tested the hypothesis that the median was equal to 25. We found that our test statistic. Was X equals 4, and we look to find the critical value that would be the case if our null hypothesis were true. And that critical value using a binomial distribution turned out to be 12 and so given that. We used our test statistic of X equals 4 donations above 25 and found that the critical value was 12. Since our test statistic 4 is less than 12, there is not sufficient evidence to reject the null hypothesis and therefore there is insufficient evidence for the claim that. The number of weekly donations is greater than 25. See you in the next video.

Key Concepts

Null Hypothesis

Sign Test

P-value

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