Here is a simple probability model for multiple-choice tests. Suppose each stude

Question

Here is a simple probability model for multiple-choice tests. Suppose each student has probability p of correctly answering a question chosen at random from a universe of possible questions. (A strong student has a higher p than a weak student.) Answers to different questions are independent. (Round your answers to four decimal places.)

(a) Stacey is a good student for whom p = 0.75. Use the Normal approximation to find the probability that Stacey scores between (and including) 70% and 80% on a 100-question test.

(b) If the test contains 250 questions, what is the probability that Stacey will score between (and including) 70% and 80%? You see that Stacey's score on the longer test is more likely to be close to her "true score."

Please clarify your answers in bold so I can see which ones are the answers better. Please also make sure they're rounded to four-decimal places before answering my Chegg question. Thanks.

Explanation / Answer

a)

mean = n*p = 100*0.75 = 75

std = sqrt(n*p*q) = sqrt(75*0.25) = 4.33
----
Binomial: P(70<= x <=80) ~ Normal:P(69.5<= x <=80.5)

z(69.5) = (69.5 - 75)/4.33 = - 1.27

z(80.5) = (80.5 - 75)/4.33 = 1.27
------
Ans: p(- 1.27 < z < 1.27) = 0.7959

b)

mean = n*p = 250*0.75 = 187.5

std = sqrt(n*p*q) = sqrt(187.5 * 0.25) = 6.85
----
Binomial: P(70<= x <=80) ~ Normal:P(69.5<= x <=80.5)

Note: 0.7 * 250 = 175 ; 0.8*250 = 200

z(69.5) = (175 - 187.5)/6.85 = - 1.82

z(80.5) = (200 - 187.5)/6.85 = 1.82
------
Ans: p(- 1.82 < z < 1.82) = 0.9312

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