Feeling Your Age A research organization conducts a survey by ...

Question

Feeling Your Age A research organization conducts a survey by randomly selecting adults and asking each, “How do you feel relative to your age?” The results are shown in the figure. (Adapted from Pew Research Center)

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a. Use a sign test to test the null hypothesis that the proportion of adults who feel older is equal to the proportion of adults who feel younger. Assign a + sign to each adult who responded “older,” assign a - sign to each adult who responded “younger,” and assign a 0 to each adult who responded “my age.” Use α = 0.05

Accepted Answer

All right, hi everyone. So, this question says, a research team surveys 100 employees and asks each, how do you feel about your current stress level compared to your colleagues? The responses are as follows. Using the sign test, examine the null hypothesis that the proportion of employees who feel more stressed equals the proportion who feel less stressed. Use alpha equals 0.05. And here we have 4 different answer choices labeled A through D. So first, notice that the claim is that the proportions of those feeling more stressed and less stressed are equal. So here, the null hypothesis. would state that P, the proportion of those that feel more stressed, is equal to P negative, the proportion of people that feel less stressed. In other words, the median difference would be equal to 0. The alternative hypothesis, on the other hand, would state simply that P positive is not equal to P negative, making this a two-tailed test. So, first, let me go back to the data set, because the first step is to eliminate any ties. So observations equal to about the same. In this case, there are 18 employees that said about the same, so those are the ones that we eliminate. The effective sample size, therefore, is equal to the original 100, subtracted by the 18 that were removed, which gives you 82. And now to find Both S positive and S negative. As positive, the number of people that reported feeling more stressed is equal to 47. And S negative is 35. This now leads you to find the critical value. So under the null hypothesis. S positive follows a binomial distribution such that N is equal to 82 and P is equal to 0.5. Now, for a two-tailed test at alpha equals 0.05, the critical bounds are determined by the probability of S positive being less than or equal to K, being less than or equals to 0.025. And by the probability of S positive being greater than or equal to 82 subtracted by K, being less than or equal to 0.025. Notice that 0.025 is alpha divided by 2. So now, using either binomial tables or software, this gives you critical values of K equals 31. And 82 subtracted by K equals 51. Therefore H knot can be rejected. If as positive, oops. If S positive is either less than or equal to 31. Or as positive is greater than or equal to 51. But in this case, S positive is in fact equal to 47, which is between our critical values. Because it's between our critical values, it does not fall in the rejection region, which means that ultimately we fail to reject. H not we fail to reject the null hypothesis. What this tells you, therefore, is that at the 5% significance level, there is not enough evidence to conclude that the proportion of employees feeling more stressed differs from the proportion feeling less stressed. So ultimately The correct answer is going to be option B in the multiple choice. And there you have it. So with that being said, thank you so very much for watching, and I hope you found this helpful.

Key Concepts

Sign Test

Null Hypothesis

Significance Level (α)

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