Please help with Statistics! Thank you! 1.A new medical test provides a false po

Question

Please help with Statistics! Thank you!

1.A new medical test provides a false positive result for Hepatitis 2% of the time. That is, a perfectly healthy subject being tested for Hepatitis will test as being infected 2% of the time. In research, the test is given to 30 healthy (not having Hepatitis) subjects. Let X be the number of subjects who test positive for the disease.

a) What is the probability that all 30 subjects will appropriately test as not being infected?

b) What are the mean and standard deviation of X?

c) To what extent do you think this is a viable test to use in the field of medicine?

2. Some people believe in the force (yes, like in Star Wars). To test if young Anakin Skywalker has the force with him, he is told that some cards that Yoda can see but Anakin cannot contain a picture of a planet, a lightsaber, a laser rifle, or a spaceship. As Yoda looks at 20 such cards in turn, Anakin tries to guess what is on the card Yoda is looking at. Of course, Anakin has a 25% chance of simply guessing correctly.

a) Verify that the count of correct guesses in 20 cards follows a binomial distribution and write the notation.

b) What is the mean number of correct guesses?

c) What is the probability that Anakin guesses all 20 cards correctly?

d) Suppose Anakin guesses correctly on 10 of the cards. What is the probability of him doing this well or better by chance? Do you think he has the force with him?

Explanation / Answer

Solution:

1. Binomial distribution is used.
P(not infected) = 100%-2% = 98% = 0.98
q = 1-p = 1-0.98 = 0.02
n = 30
P(r) = nCr*q^(n-r)*p^r
a) Required probability = P(r=30) = 30C30*0.02^0*0.98^30
= 1*1*0.5455
= 0.5455
b) Mean = np = 30*0.98 = 29.4
Standard deviation = sqrt (npq)
= sqrt (30*0.98*0.02)
= 0.7668
c) This is not a viable test.
Notice that from a), for 30 test subjects, there is almost a 0.5 probability that there will be at least one test subject with a false positive.

2. a) x~bin(20,0.25)

b) µ = np = 20*0.25 = 5

c) P[20] = 0.25^20 = 9.0949e-13

d) P[10] = 0.0139 or 1.39%
so 10 guesses correct is unlikely by chance, but not that unlikely that i'd belirve that the force is with him

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