Step 1 :The number of trials, n, is the total number of students taking the course, which is 88.
Step 2 :The probability of success, p, is the probability that a student passes the course, which is 65% or 0.65.
Step 3 :The probability of failure, q, is the probability that a student fails the course, which is 1 - p = 1 - 0.65 = 0.35.
Step 4 :So, n = 88, p = 0.65, q = 0.35.
Step 5 :The mean and standard deviation of a binomial distribution are given by the formulas: Mean, \(\mu = np\) and Standard deviation, \(\sigma = \sqrt{npq}\)
Step 6 :Substituting the given values: \(\mu = np = 88 * 0.65 = 57.2\) and \(\sigma = \sqrt{npq} = \sqrt{88 * 0.65 * 0.35} = 7.141\) (rounded to three decimal places)
Step 7 :To find the probability that at least 56 pass the course, we need to find the z-score for 56. The z-score is given by the formula: \(z = (X - \mu) / \sigma\)
Step 8 :Substituting the given values: \(z = (56 - 57.2) / 7.141 = -0.168\) (rounded to three decimal places)
Step 9 :The probability that a z-score is at least -0.168 is 1 - P(Z < -0.168) = 1 - 0.4332 = 0.5668 or 56.68%.
Step 10 :To find the probability that no more than 63 pass the course, we need to find the z-score for 63. Substituting the given values: \(z = (63 - 57.2) / 7.141 = 0.811\) (rounded to three decimal places)
Step 11 :The probability that a z-score is no more than 0.811 is P(Z < 0.811) = 0.7910 or 79.10%.