Question

7. Records show that a treatment for a rare disease is effective 70% of the time. Suppose that a random sample of 18 patients undergoes this treatment. (a) Find the probability that the treatment will be successful for (i) exactly 13 patients iii) under 10 patients(ii) at least 13 patients (iv) 10 to 15 patients inclusive (b) Find: () the mean (expected) number patients for which the treatment will be successful (ii) the variance and standard deviation

Answer 1

it is a binomial probability distribution,

and probability is given by

P(X=x) = C(n,x)*px(1-p)(n-x)

Sample size , n =    18
Probability of an event of interest, p =   0.7

1)

a)

P(x=13) = C(18,13)*0.7¹³(1-0.7)^(18-13)=0.2017

b) P(X<10) = $\sum_{x=0}^{9}$ C(n,x)*p^x(1-p)^(n-x) = 0.0596

c) P(X> = 13) = $\sum_{x=13}^{18}$ C(n,x)*p^x(1-p)^(n-x) =0.5344

d)

P(10<=X <=15) =  $\sum_{x=10}^{15}$ C(n,x)*p^x(1-p)^(n-x) =0.8805

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2)

1)Mean = np =    12.6
2)Variance = np(1-p) =    3.7800
Standard deviation = √variance =    1.9442