Suppose that in a population of students in a course with a large enrollment, th

Question

Suppose that in a population of students in a course with a large enrollment, the mark, out of 100, on a final exam is approximately distributed N (, 9). The instructor places the prior (65,1) on the unknown parameter. A sample of 10 marks is obtained as given below. 46, 68, 34, 86, 75, 56, 77, 73, 53, 64 (a) Determine the posterior mode and a 0.95-credible interval for . what does this interval tell you about the accuracy of the estimate? (b) Use the 0.95-credible interval for to test the hypothesis Ho: = 65

Explanation / Answer

Solution

Back-up Theory

Let X = score in the examination. We are given X ~ N(µ, 2), where prior value of µ is given to be 65. 2 is given to be 9 or = 3.

For Normal Distribution, mean = mode = µ.

Part (a)

Given the 10 sample values, posterior mode = sample average = X bar

= (46 + 68 + 34 + 86 + 75 + 56 + 77 + 73 + 53 + 64)/10

= 63.2 ANSWER 1

95% credible interval for µ is: {Xbar ± (/n)(Z/2)}, where n = sample size = 10. Z/2 = upper 2.5% point of N(0, 1) = 1.96 [from Standard Normal Tables]

= 63.2 ± (3/10)(1.96)

= 63.2 ± 1.859

= (61.341, 65.059) ANSWER 2

Part (b)

Since the above interval does contain 65, at 5% level of significance, it is reasonable to infer that the hypothesis H0: µ = 65 is acceptable. ANSWER

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