Question 1: (20 points)
(A) The probabilities that a bakery has a demand for 2, 3, 5, and 7 birthday cakes on any given day are 0.35, 0.41, 0.15, and 0.09, respectively.
(i) Construct a probability distribution table for the above information.
(ii) Determine the expected value and variance of demand on cakes.
(B) The probability that an American citizen holds another nationality is 28%. If 25 Americans are selected at random.
a) Using Binomial Distribution, Calculate the probability that exactly 8 persons hold another nationality?
b) What is the mean of the number of those who hold another nationality?
c) What is the variance of the number of those who hold another nationality?
Question 2 (20 points)
A Statistics department purchased 24 hand calculators from a dealer to have a supply on hand for tests for use by students who forget to bring their own. Although the department was not aware of this, three of the calculators were defective and gave incorrect answers to calculations. Students who have forgotten their own calculators are allowed to select one of the Department's (at random). Suppose that five students forgot to bring their calculators.
(Note: Use hypergeometric probability distribution to calculate probabilities)
a) What is the probability that exactly two of these students select a defective calculator?
b) What is the probability that no one (x = 0) of these students selects a defective calculator?
(10 + 10 = 20 marks)Question 3: (20 points)
(A) A worker in the automobile industry works an average of 43.7 hours per week. Assume the distribution is normal with a standard deviation of 1.6 hours.
(i) What is the probability that a randomly selected automobile worker works less than 40 hours per week?
(ii) If 15 automobile workers are randomly selected, what is the probability that the sample mean of working time is more than 45 hours per week?
(B) An airline knows from experience that the distribution of the number of suitcases that get lost each week on a certain route is normal with μ (mean) = 15.5 and σ (standard deviation) = 3.6. What is the probability that during a given week the airline will lose between 11 and 19 suitcases?
(6 + 7 + 7 = 20 marks) Question 4: (20 points)
(A) The weight of cans of vegetables is normally distributed with a mean of 1380 grams and a standard deviation of 80 grams. What is the probability that the sample mean of weight for 15 randomly selected cans is more than 1410? (10 points)
(B) The age of vehicles registered in a certain European country is normally distributed with a mean of 85 months and a standard deviation of 20 months. What is the probability that the sample mean of age for a sample of 16 vehicles is between 92 and 98 months? (10 points)