MGQ 201 WEEK 9 VICTORIA LOJACONO

Chapter 5 Question 1 A discrete probability distribution equals 1 I did this in excel to make it easier, but all you need to do is sum the given P(x) values and subtract your answer from 1.

Question 2 Notes Part A: Compute the mean Step 1: Find the total number of minutes (if it is not already given) Step 2: Find the probability by dividing the frequency of each number by the total number of Minutes. Step 3: multiply each x value by the corresponding probability you just calculated Step 4: sum all of your values in step 3 Part B: Compute the standard deviation Step 1: Calculate (x – the mean you found as the answer for part A) for each x value Step 2: square each of those values Step 3: Multiply each value in step 2 by the probability you found in step 2 of part A Step 4: sum all of your values in step 3 Step 5: take the square root of step 4 *I did this in excel to avoid doing so many calculations by hand, but you could do it the same way on paper

Question 2 *see excel for step by step answers

Question 3 Notes Expected monetary value (EMV) = [Probability1*profit for demand1] + [probability2 * profit for demand2] + [probability3 * profit for demand3] ….. Probabilities are given in the question. Profit for each level of demand needs to be calculated and then plugged in to the EMV formula. The option with the highest EMV is the best option that should be recommended to the company.

Question 3 Given: 1 st Possible Demand is 25 2 nd Possible Demand is 50 3 rd Possible Demand is 75 Probability of Demand of 25 is 30% Probability of Demand of 50 is 20% Probability of Demand of 75 is 50% Each loaf costs $2 Each loaf is sold for $5 Need to determine:  If they make 25 loaves:  Profit for each demand  EMV  If they make 50 loaves:  Profit for each demand  EMV  If they make 75 loaves:  Profit for each demand  EMV Choose the option with the highest EMV. *Detailed work for this problem is posted in Question 3 Solutions on UBLearns

Question 4 Notes Formula for binomial distribution: n and p are given in the question x is given in parts A, B, and C  When finding a probability for an “exact x” just plug that number in for x.  When finding a probability for “less than x” you need to plug every value in for x starting with 0 and ending with the value below x. Then sum the probabilities.  When finding a probability for “x or more” you need to plug every value in for x starting with x and ending with n. Then sum the probabilities.

Question 4 Part A: Plug 3 in for x, and plug the given values in for p and n

Question 4 cont. Part B: Less than 3 successes Plug 0, 1, and 2 in for x Then sum the answers Part C: 8 or more successes Plug 8, 9, and 10 in for x (no more than 10 because 10 is n) Then sum the answers *When doing these problems, do not round anything until the very end!

Question 5 Notes This question wants you to find the binomial probability using excel. The formula is: =BINOM.DIST(x, number of trials, probability of success, TRUE or FALSE) ◦Use TRUE (cumulative) if the question is asking for a probability that is “less than or equal to” x ◦Use FALSE (non cumulative) if the question is asking for a probability of an exact x Mean = np which is sample size (n) * probability of success (p) Standard deviation = n = number of trials p = probability of success q = probability of failure

Question 5 *see excel

Question 6 Notes We will be using the Poisson distribution table to solve for the answers in this question.

Question 6 – How to use the table Part A: P(x=5) Given: Probability distribution is 2.4 Part B: P(x>6) This is the same as 1 – (sum of the probabilities 0,1,2,3,4,5,6) Find each one in the table the same way. Part C: P(x<3) This is the same as the sum of the probability of 0,1,2,3 *watch your greater than/less than/equal to signs so that you don’t forget a value!

Question 7 A is the red circles B is 1 – the blue circle C is green circle

Question 8 Notes Hypergeometric distribution formula: N = population size R = number of successes in the population n = sample size x = number of successes in the sample Combination formula: Represents the number of combinations of n objects selected x at a time

Question 8 – Part A Step 1: Plug given values into hypergeometric formula Step 2: Use combination formula to find the 3 combinations that are left in the hypergeometric formula (or use the nCr button on your calculator!) Step 1 Step 2

Question 8 – Part C Step 1: When the probability is x<1, plug every value below and including 1 into hypergeometric formula (you will have two formulas, one with 0 as x and the other with 1 as x) Step 2: Use combination formula to find the 3 combinations that are left in each hypergeometric formula Step 3: Add the two probabilities together

Question 8 – Part D Calculate the mean and standard deviation using these formulas and plugging in given values:

Question 9 Notes Step 1: Define the variables N = population size R = number of successes in the population n = sample size x = number of successes in the sample Step 2: Plug them in to the hypergeometric formula Step 3: Use the combination formula to find the 3 combinations within the hypergeometric formula

Question 9 Step 1: Step 2:Step 3: N = population size (8+7=15) R = number of successes in the population (number of blue balls in the urn) n = sample size (number of balls selected) x = number of successes in the sample (of the balls selected, 1 will be blue)

Question 10 Notes For Question 10, you can use excel to calculate the same hypergeometric probabilities in a much easier way. Step 1: Define your variables: N = population size R = number of successes in the population n = sample size x = number of successes in the sample Step 2: Use the formula in excel: =HYPGEOM.DIST(x,n,R,N,false)

Question 10 *This problem is the same as the last one, but you can do it in excel. See excel sheet for answers

Reminders  Stats quiz 5 due 11/2  Simulations due 12/15