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

Question

Performing a Sign Test In Exercises 7–22, (b) find the critical value,

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)

Hourly Wages A labor organization claims that the median hourly wage of computer systems analysts is $41.93. In a random sample of 45 computer systems analysts, 18 earn less than $41.93 per hour, 25 earn more than $41.93 per hour, and 2 earn $41.93 per hour. At , can you reject the labor organization’s claim? (Adapted from U.S. Bureau of Labor Statistics)

Accepted Answer

Hello and welcome back, everybody. The next problem says A courier company claims that the median delivery time for local packages is 30 minutes. In a random sample of 15 deliveries, the recorded times in minutes were 28, 31, 30, 29, 32, 35, 30, 33, 29, 36, 34, 30, 37, 38, 30. And then the problem is in two parts. Number 1, identify the claim and state the null hypothesis and alternative hypothesis. And number 2, at the 0.05 significance level, can you reject the company's claim? Interpret the decision. So our claim here is about the median delivery time. So our claim Is that the median. It is 30 minutes. So median is equal to 30. So, note that we have an equality statement here. So, we will set this claim as our null hypothesis, or null hypothesis will be that the median is equal to 30. And the alternative hypothesis would then be that the median. Does not equal 30. So we have done part one here. And then for part two, we're told the significance level, so alpha equals 0.05. And then it says, can we reject the company's claim. So, we have a sample size of less than 30, it's 15 deliveries. And we have just 2 results since we're talking about a median. The result could be above or below the median. So we're going to do a sign test. So, how do we do that? Well, first we look at our samples, and where we've said the median is equal to 30. So, in our sample batch, we're going to remove all ties. So, the media is 30, remove every 30 we see in that batch. I'm going to use a red X to cross them out. So I see 130. A 2nd 30, and a 3rd value of 30. 1 more at the tail end 4. So, 4 of our samples. Are eliminated. So what does that affect? It changes our sample numbers. Instead of N equaling 15, we have 15 deliveries. RN will be. 15 minus 4. The number of zeros, the number that were the same as the median, so that's going to equal 11. So we've got in We've got Alpha. So, next to the scientist, we've eliminated the zeros, and we need to count up how many positive and how many negative values. So, we'll go ahead and assign our positive. Remember, it doesn't really matter which we assign as which, as those values, where the time it took to deliver, is greater than 30 minutes. So let's look for those. We have a 31. 32, 35, 33, 36, 34, 37, and 38. So, 12345678. So there were 8 of those samples. All labels negative, the ones. Where the time was less than 30. And we'll go ahead and underline that in red, so we've got 28, 29. 229. And that seems to be it. And we just get 3 of those with the delivery time below 30. So, our N here is going to be the sum of our positive and negative amounts. So, N, the new N, will equal 8 and 311. So, now we have our alpha, we have our N, so we can use the table. So we can use these to look in the table for critical values for scientists. So, N11, and then we recall that this is going to be a two-tailed scientist. And when we do that, we get a critical value. Of one. Now we need to compare our test statistic to that critical value. So, what is the test statistic in this case? We have a two-tailed scientist. So, our test statistic X. So, we look at our number of positive values was 8, number of negative values was 3. The smaller of those two numbers. is 3, and that will equal X and be our test statistic. So we know to use that sign test then, we, we would reject the null hypothesis, when our test statistic X is less than or equal to our critical value. Again, we have R X equal to 3. Which is greater than that critical value of one. So we fail. To reject the null hypothesis, so we would conclude that at this. 0.05 significance level, we have insufficient evidence to reject. That null hypothesis. So again, we identified our claim that the median was 30. With the null hypothesis that the median is equal to 30. The alternative hypothesis that means is not equal to 30. And then we were asked that at the 0.05 significance level, can we reject the company's claim? And the answer is, we failed to reject that claim. Because when we calculated our critical value, and received a critical value of 1, we compare our test statistic to that number and reject the null hypothesis when X is less than or equal to. Oops, less than or equal to. Our critical value. So since our test statistic was 3, which is greater than that critical value of 1, we fail to reject the hypothesis. So we are not able To reject the company's claim, as that was our null hypothesis. See you in the next video.

Key Concepts

Hypothesis Testing

Critical Value

Test Statistic

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