Experience has shown that 30% of all persons afflicted by a certain illness reco

Question

Experience has shown that 30% of all persons afflicted by a certain illness recover on their own. A drug company has developed a new medication. Ten people with the illness were selected at random and received the medication; nine recovered shortly thereafter. Define X the number of patients that recover if the medication is absolutely worthless. Find the following: a) The probability distribution function of X b) The probability that exactly nine of the ten recover c) The probability that at least nine of the ten recover d) The probability that between two and four (inclusive) recover e) The expected number of patients that would recover if the medication is worthless. fGiven that we administered the medication to 10 individuals and observed 9 recoveries, what can you conclude about the efficacy of the new medication? Explain based on your prior calculations.

Explanation / Answer

p = 0.3

n = 10

a) P(X = x) = 10Cx * 0.3x * 0.710-x

b) P(X = 9) = 10C9 * 0.39 * 0.71 = 0.0001

c) P(X > 9) = P(X = 9) + P(X = 10)

= 10C9 * 0.39 * 0.71 + 10C10 * 0.310 * 0.70

= 0.0001

d) P(2 < X < 4) = P(X = 2) + P(X = 3) + P(X = 4)

= 10C2 * 0.32 * 0.78 + 10C3 * 0.33 * 0.77 + 10C4 * 0.34 * 0.76

= 0.2335 + 0.2668 + 0.2001

= 0.7004

e) E(X) = n * p = 10 * 0.3 = 3

f) Since P(X = 9) is less than 0.05, this is very unusual

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