15. (5 points) Suppose we take a random sample of size 100 from a normal populat

Question

15. (5 points) Suppose we take a random sample of size 100 from a normal population with uni nown mean but known standard deviation: 40. We plan to test His-85 versus Ha: > 85. Our procedure will be to observe the sample and then reject Ho if xba 90. A What is the probability of rejecting the nullif in fact B. What is 957 s the probability of type Il error and what is the power of the test if in fact H 95 16. (Bonus 10 points) A teacher and six of his students are in the classroom. The teacher is an Aquarius. A What is the probability that at least one of the students is also Aquarius? B. What is the probability that at least two of the seven people in the classroom share the same zodiac sign?

Explanation / Answer

(16)

There are 12 zodiac signs.

So the probability that someone is Aquarius is equal to 1/12.

(a)

Modelling this as a Binomial distribution with following parameters:

n = 6, p = 1/12

P(atleast one of the students is an Aquarius) = 1 - P(none of the students is an Aquarius)

So,

P(none of the students is an Aquarius) = 7C0*(1/12)0*(1-(1/12))12-0 = 0.352

So,

P(atleast one of the students is an Aquarius) = 1-0.352 = 0.648

(b)

P(at least two of the seven people are Aquarius) = 1 - P(only one of the seven people is Aquarius)

P(only one of the seven people is Aquarius) = P(none of the students is an Aquarius) = 0.352

So,

P(at least two of the seven people are Aquarius) = 1-0.352 = 0.648

Hope this helps !

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