Question

Medical treatment will cure about 87% of all people who suffer from a certain eye disorder. Suppose a large medical clinic treats 57 people with this disorder. Let r be a random variable that represents the number of people that will recover. The clinic wants a probability distribution for r. Use the normal approximation to the binomial distribution.

1. thachthao
Answers to all the questions using binomial distribution are shown below.

### What is Binomial distribution?

• The binomial distribution is a discrete probability distribution that indicates the likelihood of success in a replacement experiment.
• This is in contrast to the Hypergeometric distribution, which provides the likelihood of success in an experiment conducted without replacement.
To answer all the questions using binomial distribution:
(A) The binomial distribution will be roughly normally distributed if the following conditions are satisfied:
• np > 5
• n(1−p ) > 5
Given that;
• n = 57
• p = 0.87
Test whether the conditions are satisfied:
• np:57 × .87>5
• n(1-p) : 57(1−.87)>5
Since the conditions are satisfied, the distribution is approximately normal.
(B) Calculate mean and standard deviation of binomial distribution:
• E(x)=μ=np
• = 57×.87
• = 49.59
• σ =√np(1−p)
• =√57×.87(1−.87)
• = 2.539
Now use normal approximation:
• P(r≤47) = P(X−μ/σ≤47−49.59/2.539)
• = P(z≤−1.02)
• = 0.1539
(C)
• P(47 ≤ r ≤ 55) = P(47 − 49.59 / 2.539 ≤ X – μ / σ ≤ 55 − 49.59 / 2.539)
• = P(−1.02 ≤ z ≤ 2.13)
• = P(z < 2.13)−P(z < −1.02)
• = 0.9834−0.1539
• = 0.8295
Therefore, answer to all the questions using binomial distribution are shown below.
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The complete question is given below:
Medical treatment will cure about  87% of all people who suffer from a certain eye disorder. Suppose a large medical clinic treats 57 people with this disorder. Let r be a random variable that represents the number of people that will recover. The clinic wants a probability distribution for r.
A. Write a brief but complete description in which you explain why the normal approximation to the binomial would apply. Are the assumptions satisfied? Explain.
b. Estimate P(r≤47).
c. Estimate P(47≤r≤55).