Math 13 Statistics Fall 2014 Midterm 2 Review Problems Due on the day of the midterm (Friday, October 3, 2014 at 6 p.m. in N12) PRINT NAME (ALL UPPERCASE): Problem 1: A couple wants to have three babies for the next three years, only one baby per year (either boy or girl) and no other possibility. Create a sample space, i.e. collection of all simple events: Find the probability that they will have two girls and a boy. Find the probability that they will have at least one girl. Find the probability that they will have no more than two boys. Find the probability that they will have no girls. Find the probability that they will have between one and three girls. Problem 2: You need to answer two questions; one is true/false and the other is a multiple choice question with 6 choices. Create the sample space: Find the following probabilities: Getting both questions right: Getting both wrong: Getting at most two wrong: Getting at most two right: Getting at least one right:
Problem 3: A box contains 10 black socks and 10 white socks. If you close your eyes and pick socks, then how many socks do you have to pick in order make sure that you have a pair of the same color? Find the probability of picking 3 black socks in a row without replacement: Find the probability of picking 2 blacks and a white sock in a row without replacement: Problem 4: Quality Control: As a quality control manager in a clothing company you randomly select 5 shirts from a collection of 2000 shirts that just came to your company from Bangladesh. You will reject all the shirts of if you find at least one faulty shirt. It is assumed that there are 20 faulty shirts in the lot of 2000 shirts. Find the probability of accepting all the shirts in this lot. Problem 5: A student takes a multiple-choice test. Each question has 5 different choices and there are 5 different questions. Find the probability that the student gets at least one question right by pure guessing. Problem 6: An access code has 6 characters. First four are digits and the last two are alphabets which are case sensitive. A thief trying to break this code has a probability of success: Problem 7: You have the option of buying one car from 5 different types of cars and you may pick one insurance from 4 different choices. What are total number of ways you can have the car and insurance combination?
Problem 8: A student committee consists of 13 members. They need to elect a president, a vice president, and a treasurer. How many different ways this can be accomplished? Problem: Permutation and combination: What are your chances of winning the Mega Millions Lottery? You pick 5 numbers from 1 to 56 without replacement and 1 number from 1 to 46. Problem 9: Age discrimination: Among 13 managers the company laid off 3 oldest managers. Do you think there was discrimination involved in the process based on your calculations?
Problem 10: Your chances of passing the statistics class is 80% and your chances of passing another class that you are currently taking is 70%. The probability that you would pass both of the classes is 65%. What is the probability that you would pass at least one of the classes? Problem 11: Does the following form a discrete probability distribution? Explain. 1 Px ( ) = 4 x = {1, 2, 4,8,16} Problem 12: Assume that you are investing $10,000 in one bond. There are two types of bonds available. The first bond gives you a 7% return with a default rate of 3% and the second bond gives you a return of 9% with a default rate of 5%. Which one these bonds would you consider for investing your $10,000 assuming that you want to maximize your profit.
Problem 13: A new drug named CURAIDS that is 60% effective in extending the average life of an AIDS patient by twenty years. Five randomly selected AIDS patients from Africa are treated with this new drug. Answer the following questions based on the above information. (a) Show that the above situation satisfies all four criteria for the Binomial probability distribution. (b) (10 points) Fill in the probabilities in the following table. Show your calculations. x 0 1 2 3 4 5 P(x) (c) What is the probability that no more than 4 patients are cured? Use results from part (b), do not do the calculations again. (d) Find the probability that more than 2 patients or less than or equal to 5 patients are cured. Use results from part (b), do not do the calculations again. (e) Find the probability that at least 4 patients are cured. Use results from part (b), do not do the calculations again. (f) Find probability that less than 2 or more than 3 patients are cured. Use results from part (b), do not do the calculations again.
Problem 14: In a city named Dhaka there were 125 drug related crimes over one year period. Find the probability that on a given day there will be exactly 3 drug related crimes in that city. Use Poisson distribution. Explain the requirements for Poisson distribution. Normal Distribution x µ x x z = (For population) z = σ s (For sample) Problem 15: Find the z-scores for the following values if the mean income in a city is $35,000 and standard deviation is $7,000 a) Find the z-score for someone who makes $38,500 and find the associated probability from the standard normal distribution table
b) Find the z-score for someone who makes $30,500 and find the associated probability from the standard normal distribution table c) Find the corresponding income for an individual if that person s z-score is -1.35 d) Find the corresponding income for an individual if that person s z-score is 0.75
Problem 16: Find the z-scores associated with the following probabilities and then find the income of a person with that z-score if the mean income is $35,000 and standard deviation is $7,000 a) The probability is 0.05 b) The probability is 0.95 c) The probability is 0.75 d) The probability is 0.35
Problem 17: If a sample is taken from a population that is normally distributed with average income $35,000 and standard deviation is $7,000 then find the following: a) z-score for the top 3% earners and then convert it to actual income b) z-score for the bottom 3% earners and then convert it to actual income c) z-scores for the middle 50% of earners and then convert those to actual income Problem 18: The average income in Dhaka, the capital of Bangladesh is $1,500 with $300 standard deviation. Find the following probabilities for a randomly selected person from Dhaka: a) Find the probability that the person makes less than $1000
b) Find the probability that the person makes more than $1,700 c) Find the probability that the person makes between $1,000 and $1,700 d) Find the probability that the person makes exactly $1,500
Problem 19: The average usable time for a smartphone is approximately 36 months and standard deviation is 6 months. If you want to purchase an insurance policy so that there is 2% chance that your phone will NOT be fixed if it is broken. What should be the length of time for which you should buy the warranty? Assume normal distribution.