Consider the History Quiz Exercise from Week 7. If Mary, a student in the Histor

Question

Consider the History Quiz Exercise from Week 7. If Mary, a student in the History class, answered tour out of the six questions on the quiz correctly, would you believe that she just guessed at the answers? Apply the principle of modus tollens to explain why you would think that Mary studied and did not just guess on all six questions. By applying the principle of modus tollens to whether Mary guessed or not. what null and alternative hypotheses might you be implicitly testing? Write out these two hypotheses to make them explicit. A psychologist gives a verbal aptitude test to a random sample of 49 Oakland third graders. Assume that nationally on this test, third graders score on average 80 points with a standard deviation of 17. If Oakland third graders are like everyone else, what is the probability that the 49 Oakland third graders in the psychologist's sample will have a mean score of 85 or higher? On the verbal aptitude test, the 49 tested Oakland third graders actually attain an average score of 85 with a standard deviation of 17. Based on this result, which of the following two hypotheses would you have greater confidence to assert for the population of all Oakland third graders: H_0: Oakland third graders have verbal aptitudes scores, on average, equal to or lower than children nationally. H_1 Oakland third graders have verbal aptitudes scores, on average, higher than children nationally. Applying the principle of modus tollens. justify your answer.

Explanation / Answer

Exercise 1

According to modus principle, Mary should answer half of the questions correctly on an average if she just prepared. If she is prepared then it would be more than half of the questions being answered correctly by Mary. For the given scenario, there are total six questions and Mary answered four correctly. This means we can believe that she is prepared for exam. Let p be the proportion of correct answers. The null and alternative hypothesis for this test is given as below:

H0: p 0.5 V/s Ha: p > 0.5

Exercise 2

We have to find P(Xbar>85)

P(Xbar>85) = 1 – P(Xbar<85)

Z = (Xbar - µ)/[/sqrt(n)]

Z = (85 – 80)/[17/sqrt(49)]

Z = 5/[17/7] = 2.058824

Z = 2.058824

P(Xbar<85) = P(Z< 2.058824) = 0.980244

P(Xbar>85) = 1 – P(Xbar<85)

P(Xbar>85) = 1 – 0.980244

P(Xbar>85) = 0.019756

Required probability = 0.019756

Exercise 3

For the given scenario, the alternative hypothesis have greater confidence to assert for the population of all Oakland third graders because we have the sample mean 85 which is greater than the population mean 80.

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