The National Bureau of Economic Research (NBER) is a private, ...

Question

The National Bureau of Economic Research (NBER) is a private, nonprofit, nonpartisan research organization. The NBER provides information for better understanding of how the U.S. economy works. Researchers at the NBER concentrate on four types of empirical research: developing new statistical measurements, estimating quantitative models of economic behavior, assessing the effects of public policies on the U.S. economy, and projecting the effects of alternative policy proposals.One of the NBER’s interests is the median income of people in different regions of the United States. The table at the right shows the annual incomes (in dollars) of a random sample of people (15 years and over) in a recent year in four U.S. regions: Northeast, Midwest, South, and West.

In Exercises 1–5, refer to the annual incomes of people in the table. Use for all tests.
Use technology to perform a sign test to test the claim that the median annual income in the Midwest is greater than \$30,000.

Table displaying annual incomes in dollars for people aged 15 and over across four U.S. regions: Northeast, Midwest, South, and West.

In Exercises 1–5, refer to the annual incomes of people in the table. Use for all tests.

Use technology to perform a sign test to test the claim that the median annual income in the Midwest is greater than $30,000.

Accepted Answer

Hi everyone. Welcome back. Let's take a look at our next problem. A financial analyst wants to determine if the median annual bonus in a certain industry is greater than $5500. The following are the annual bonuses in dollars for a random sample of 9 employees. And then we have 9 values here that we'll come back to and read in a minute. And then our problem says use a sign test at alpha equals 0.05, to test the claim that the median bonus is greater than $5500. Our answer choices are A. There is sufficient evidence to support the claim that the median bonus is greater than $5500. There is insufficient evidence to support the claim that the median bonus is greater than $5500. The test is inconclusive, or D, the sample size is too small to test. Well, we can rule out choice D right away. We do have a small sample, and equals 9. This is too small to assume a normal distribution, but with the sine test, we can use a binomial. Distribution, since we have two possible results with the sign test, positive or negative. So, we can still test this even with that small sample size. So we start with our null and alternative hypotheses. When we're talking about a median value. Then we We are going to look at probabilities because if we set our null hypothesis. As the median. Being equal to 5500. When we use the sign test, we know that there's an equal probability, if the median, excuse me, if the null hypothesis is true, an equal probability of getting a result above or below that median, a positive or negative result. So then our alternative hypothesis is our claim. That the median is greater than 5500. So we'll be looking at a right tailed test for this. So, first we're going to sign our positive and negative values for that sign test. So we'll look at our sample values, and what we'll go ahead and do, we'd like to assign a positive value if our value. is greater than 5500. And a negative value if the value is. Less than 5500. And we'll accomplish that by taking our value and subtracting 5500. And that will just, that will give us positive and and positive and negative, excuse me, that will line up with this. Now, if there were any exactly equal to 5500, we actually exclude those from our sample. So, let's go ahead and look at these values. They are 5200, so we give that a negative sign, 5800, 5600, 5400 negative, 5700, 5900, 5300 negative, 5600 positive, and 5800 positive. So none of our values are zeros, none of them are equal to 5500. So we go ahead and we'll go ahead and count how many positive results we had. So and highlight them in blue. So we've got 12. 3456 positive results. So, positive results, we have 6. And then, we don't exclude anything, so since our sample size is 9, we must have 3 negative results. Now, for a right-tailed test like this, we know that we had 9, or excuse me, 6 successes, 6. Numbers that We're above that 5500 value, and that's what we're testing for. So, what we want to say is, we're going to look for the probability of getting 6 or more successes. So, what we had, or something even more extreme to test that null hypothesis. So we want the probability. That X our number of successes. Is greater than or equal to 6. Remembering that our sample size is 9. So with a binomial distribution, that formula, we have the sum of. And then it's a combination. 9, choose K. So, how many ways can you get X successes in a group of 9 or a sample size of 9? And then from K equals 6. Up through k equals 9 and then multiplied by. And it will be P to the x power multiplied by Q to the n minus x. Where P and Q, P being the possibility of success, and Q being the possibility of failure. But again, because we're talking about a median, P and Q are equal to each other, and both are 0.5. So, that means that this formula is, again, the sum of 92 K. Multiplied by PNQ both being 0.5, and then we can add their exponents, so X plus N minus X, 0.5 to the end power, so 0.5 to the ninth power. And that is from K equaling 6 through K equals 9. So, we'll do a calculation of our combinations here. So, we start with 92 6. And remembering how the combinations work, that's going to give us 9 equals 9 factorial on the numerator, divided by, and then we'll have. 9 minus 6, so 9 or 3 factorial multiplied by. 9 factorial, excuse me, 6 factorial. So, have a moment to do that calculation. You will get 84. I won't write out that formula for all of these, it's the same way. You can do that and you will get 4 92 7. That's going to equal 36. 9 choose 8 will equal 9. And 9 choose 9 of course will equal 1. So, we'll go ahead and plug that into our formula. So we have the sum of. All of those combinations, so 84, so in parentheses 84. Add 36, add 9, add 1. So that whole sum in parentheses, that's going to be multiplied by 0.5 to the ninth power. Again, that's our probability that X is greater than our equal to 6. So when we do that, to calculate that probability. We're going to get approximately. 0.254. So now we just need to compare that probability of getting 6 or more successes to our alpha. So we'd say that 0.254 is greater than 0.05, which is our alpha value. So we fail to reject the null hypothesis. We would need our probability to be less than. That 0.05. Value to show that our sample values are divergent enough from that medium. So, we, our answer choice here is choice B. There is insufficient evidence to support the claim that the median bonus is greater than $5500 because our calculated probability. was greater than our alpha value. So we're going to go ahead and cross out choice A, or sufficient evidence, we would have had to have a calculated probability of less than 0.05. And cross out choice C because our test is conclusive. So again, we were trying to determine if the median annual bonus was greater than 5500, given this random sample of 9 employees. We used a sign test with alpha equals 0.05, using a binomial distribution. And our calculated probability was greater than the alpha value. Therefore, our answer is choice B. There is insufficient evidence to support the claim that the median bonus is greater than $5500. See you in the next video.

Key Concepts

Median Income

Sign Test

Hypothesis Testing

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