Answers

1. 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.

   Know more about Binomial distribution here:

   https://brainly.com/question/9325204

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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).

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