Contacting Parents A research organization conducts a survey b...

Question

Contacting Parents A research organization conducts a survey by randomly selecting adults and asking each, “How frequently do you contact your parents by phone?” The results are shown in the figure. (Adapted from Pew Research Center)

Pie chart showing survey results: 12 adults contact parents weekly, 8 daily, and 6 other.

a. Use a sign test to test the null hypothesis that the proportion of adults who contact their parents by phone weekly is equal to the proportion of adults who contact their parents by phone daily. Assign a + sign to each adult who responded “weekly,” assign a - sign to each adult who responded “daily,” and assign a 0 to each adult who responded “other.” Use α = 0.05

Accepted Answer

Hi, everybody, and welcome to our next question. It says a health study asked participants, do you exercise in the morning or evening? Responses are morning, evening, and never. Use a sign test to test if the proportion exercising in the morning equals the proportion exercising in the evening. A sign positive to morning, negative to evening, and zero to never. Suppose there are 12 morning, 17 evening, and 6 never responses. Use alpha equals 0.05. What is the p value? A 0.3170, B 0.4582, C 0.2291, or D 0.5000. So, we know we're going to use a sign test here, and as always, we start with our formulating our hypotheses. So, we've got our no hypothesis, we are told to. It would be that the proportion exercising in the morning equals the proportion exercising in the evening, since we're supposed to test if that is the case. So that would mean that the probability of success, the probability of big X is equal to 0.5, since we're testing if the two are equally likely. So then our alternative hypothesis in that case, Is just that the probability of X is not equal, make that a little bigger, not equal to 0.5. So, we're going to look at a two-tailed test for that probability. Well, to find a P value, and using the sine test, we're going to need our N, so our number, our sample number. And our test statistic. So when we calculate n. Notice that we had originally 3 answers, morning, evening, or never. But for a sign test, we exclude the zeros. So for N, all right, exclude. Put in quotations, zero response. Since we don't mean don't exclude any, but we mean exclude any response that was 0. We just have our morning and evening responses. So we had 12 morning responses, so N will equal 12. And then add to that our 17 evening responses, so our N will equal 29. And now for a test statistic of X, our number of successes, when it comes to a sign test, we look at the smaller number of either the positive or negative results. So, our results were 12 and 17. So, our test statistic. X is equal to the smaller of these numbers or 12 So that's what we will treat as our number of successes when we do the equation. So to know how we're going to use the binomial calculator to figure this out, we need to think about again what we're looking for, and we're looking. In that P value, We're looking for the probability of observing the test statistic, so 12, or a more extreme number. If the null hypothesis is true. So that means that how we have to calculate that P value is that P is equal to, and remember we have to multiply by 2 because we have a two-tailed test, we have, it's the statistic is what we observe or more extreme. So, 2, since our distribution will be symmetrical, multiplied by the probability, and we can phrase this as in parentheses, the word binomial. And then parentheses again, our N is 29, 0.5. Our probability of X and we're looking for the probability. That this is less than or equal to 12, since 12 is our minimum number. And we use that calculator since this involves summing up the probabilities of all the numbers from 0 to 12. So, if you're putting that into a binomial calculator, you're going to say the minimum X is 0 and the maximum X is 12. So, that will give the p-value equal to 2 multiplied by, and when we put this probability into our binomial calculator, that's going to give us a result. You can pause here if you want to do that yourself. Of approximately 0.2291. Again, we don't want to forget to multiply that by 2, and that gives us our P value of 0.4582. Again, that 2 being because we have a two-tailed test. So we see that that corresponds to answer B. And again If we have this survey, and we've got, we're looking at testing if an equal number of people exercise in the morning or evening, and we have people answering the survey as morning, evening, or never. We're going to use a sign test, because our never responses were assigned to zero, we don't include them in our sample number. Our sample number number comes from a number of positive and negative responses. So we've got a sample number. We know our probability of our null hypothesis is 0.5, equally likely. And so our alternative hypothesis, we're looking for a two-tailed test, since the alternative hypothesis is that our probability is not equal to 0.5. So we use. Our binomial calculator with that X of 12, since that's the smaller of our two numbers. And we multiply by 2, since we have a two-tailed test, and that gives us a P value of choice B, 0.4582. See you in the next video.

Key Concepts

Sign Test

Null Hypothesis

Significance Level (α)

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