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.

Sales Prices of Homes A real estate agent claims that the median sales price of new privately owned one-family homes sold in Newark, New Jersey, in a recent month is $500,000 or less. The sales prices (in dollars) of 10 randomly selected homes are listed below. At , is there enough evidence to reject the agent’s claim? (Adapted from National Association of Realtors)

Table displaying sales prices of 10 randomly selected homes in Newark, New Jersey, ranging from $200,000 to $825,000.

Accepted Answer

Hi everybody, glad to have you back. Our next problem says, a nutritionist claims that the median daily calorie intake for adults in a certain city is 2200 calories or less. The daily calorie intakes and calories of 12 randomly selected adults are. And then we have a list of 12 numbers. We'll come back to that when we look at each of these numbers individually. And then our question continues at alpha equals 0.05, is there sufficient evidence to reject the nutritionists claim? A, there is sufficient evidence to reject the claim. B, there is sufficient evidence to support that the median is greater than 2200 calories. There is sufficient evidence to support that the median is less than 2200 calories, or D insufficient data. So, let's look at what we have here. We have a problem about a median, and what the median of a given sample is. So, with median problems, We will be looking at the sign test. Because we know that with a given median there is a 50% probability that the Calorie intake will be below that median and a 50% possibility that the calorie intake will be above that median. So we can assign negative and positive signs to those two results. So, let's first look at our numbers that we didn't read out yet, and we'll assign a plus for above the median. A negative for below the median, and then a 0 for anything that ties that's equal to the median. So our claim is that the median is 2200 calories or less. So this median 2200 So the numbers we have, I'm going to do this in Red 2100, it's below 2250, above. 2300 above 2150 below. 2400 above, 2200, we give that a 0. 2350 above 2180 below. 2250 above 2100 below. 2450 above. 2250 above. So, we've looked at our data and values that we have. We will start by looking at our null hypothesis. So no hypothesis would be well. We have a claim that the median is 2200 calories or less, so we can say that the null hypothesis is that the median. Is less than or equal to 2200. And given that, we would say that our alternative hypothesis is that the median is greater than 2200. So now, looking at the number of our scores, we're going to exclude any ties, which is one of them, and we'll say how many positive results. So, we have positive results. 1234567. Positive results are 7, negative results then are 1234. We leave out the tie, and so our sample size N will equal 11. So add together 7 and 4. Now, in the science test here to get our test statistic. We know that our N value of 11 is less than 25, so our test statistic. will be that X is equal to. The smaller of our 2 number of results, positive or negative. So, we had 7 and 4 as the numbers of our results. So, X will equal 4, our test statistic. I just jotted down smaller number of results to remind us. So, we have our test statistic, our N, and we know that our alpha is 0.05. So let's look up. Our critical value recalling that since we are looking at Our alternative hypothesis being that the median is greater than 2200. That means we'll be looking at a right tailed. Scientist. So then we go to our binomial distribution table. We have our sample size of 11. And our alpha value of 0.0.0.05. So we get two possible critical values at that N equals 11, which are 2 and 9 from the table, but again, we're looking at a right tailed scientist. So our critical value. will be 9, and again, that's from the table. So now we just need to compare our test statistic with our critical value. So we have. X equaling 4, which is less than 9. And since that's the case, we are going to reject. The null hypothesis. Let's go ahead and highlight that. So our answer choice will be choice A. There's sufficient evidence to reject the claim. Now we have these other choices, it's important to note they're phrased in the language there's sufficient evidence to support. That the median is greater than or less than 2200 calories. But this isn't how you would express your results from testing hypothesis. Our claim is the null hypothesis, and so the most we can do is say that we have sufficient evidence to reject, or insufficient evidence to reject. And then we did not have choice D insufficient data, we did have enough data to calculate. So once more, we have a claim about the medium daily calorie intake being 2200 calories or less, and then we have a sample of 12 adults with their daily intakes, and we're looking for alpha equals 0.05. Is there sufficient evidence to reject the nutritionists claim? We looked at the critical value, found that our test statistic is less than the critical value. So, we will reject the null hypothesis. So, the answer is choice A, there is sufficient evidence to reject the claim. See you in the next video.

Key Concepts

Null and Alternative Hypothesis

Sign Test

Critical Value and Test Statistic

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