PROBLEM 1 (8 points):

What is the expected value of a lottery ticket where there are only four chances in a million of winning the grand prize of $20 Million?

PROBLEM 2 (8 points):

You have been offered a chance to purchase a lottery ticket with a 10% chance of making $1000; 20% chance of making $100; and 70% chance of making $0. The price of the ticket is $20.

Should you buy it? Why?

PROBLEM 3 (9 points):

You have been offered a business deal. You estimate that there is a 1% chance of making $40,000; 14% chance of making $10,000; 85% chance of making $100. How much should you be willing to pay for this deal?

PROBLEM 4 (25 points):

Suppose SUNY professors have an average life, normally distributed, of 75 years with a population standard deviation of 8 years.

a) What percent of SUNY professors will live more than 80 years?

b) What percent of SUNY professors will not make it past the age of 60?

c) Calculate the 92^nd percentile.

d) Calculate the 5^th percentile.

e) What proportion of SUNY professors will live between 75 and 85 years?

PROBLEM 5 (20 points)

The average hourly wage of plumbers is normally distributed with a population mean of $34.00 and a population standard deviation of $6.00. Calculate the following:

The proportion of plumbers earning between $18 and $22

The proportion of plumbers earning more than $38

The proportion of plumbers earning less than $15

The 75th percentile

PROBLEM 6 (10 points)

Scores of high school seniors taking the English Regents examination in New York State are normally distributed with a mean of 75 and a standard deviation of 8. Find the probability that a randomly selected high school senior will have a score between 75 and 80?

PROBLEM 7 (20 points)

The average individual monthly spending in US for paging and messaging services is $15. If the standard deviation is $2.5 and the amounts are normally distributed:

a) What is the probability that a randomly user pays more than $9.00?

b) What is the probability that a user pays below $12.00?

c) Compute the 70% percentile.

Bus310 - Homework expectedvalue-normaldistribution HOMEWORK 4 EXPECTED VALUE AND NORMAL DISTRIBUTIONS Your Name: ________________________________ EXPECTED VALUES: E(X) = XiP(Xi) PROBLEM 1 (8 points): What is the expected value of a lottery ticket where there are only four chances in a million of winning the grand prize of $20 Million? PROBLEM 2 (8 points): You have been offered a chance to purchase a lottery ticket with a 10% chance of making $1000; 20% chance of making $100; and 70% chance of making $0. The price of the ticket is $20. Should you buy it? Why? PROBLEM 3 (9 points): You have been offered a business deal. You estimate that there is a 1% chance of making $40,000; 14% chance of making $10,000; 85% chance of making $100. How much should you be willing to pay for this deal? Bus310 - Homework expectedvalue-normaldistribution (SAMPLING) NORMAL DISTRIBUTIONS General thoughts for this problem set: Any normal distribution can be converted into a standard normal distribution by transforming the normal random variable into the standard normal random variable: Z= X = the mean and = is the standard deviation. For each question you need to ask yourself if you're looking for X or you're looking for Z: You're looking for X if you're given a percentile and asked to come up with a concrete number (number of years or score on a test) You're looking for Z if you're given a concrete number (like a score or number of years) and asked to find out a probability or a percentile. Based on the Z value from the table you can find the area under the curve. Before solving the homework, again, carefully review the lecture notes. It is important to also understand why you are doing the specific computations. The next step would be for you to check and understand the solved problems from the lecture notes and the solution of the following problem. It is always useful to draw the picture first. I strongly suggest you to use the z-table of the Lecture Notes 4C. PROBLEM : Suppose that New York State high school average scores, for students who graduate, are normally distributed with a population mean of 70 and a population standard deviation of 13. a) The \"middle\" 95% of all NYS high school students have average scores between ______ and ________? b) What proportion of NYS high school students have average scores between 60 and 75? c) Calculate the 14th percentile. d) Calculate the 92nd percentile. Bus310 - Homework expectedvalue-normaldistribution SOLUTION a) The \"middle\" 95% of all NYS high school students have average scores between ___ and ___ ? .4750 .4750 ? 70 ? HS average ________________________________ -1.96 0 1.96 Z If you want to consider the middle 95%, you will have to take 2.5% from the left tail and 2.5% from the right tail. On each direction, from the middle (z = 0) there is a distance of 50% - 2.5% = 0.4750. From the chart, to an area of .4750 corresponds a z =1.96 X = +( Z)x = 70 + ( 1.96) x13 = 70 1.96 (13) 70 25.48 ANS: 95% of all NYS high school students have average scores between 44.52 and 95.48. b) What proportion of NYS high school students has average scores between 60 and 75? .2794 .1480 60 70 75 HS average ________________________________ -.77 0 .38 Z Z1 = (60 - 70) / 13 = -0.77 (it corresponds to the area under the curve from the mean value to 60; the area on the left of the mean; it is the probability of having an average value between 60 and 70) Z2 = (75 - 70) / 13 = .38 (it corresponds to the area under the curve from the mean value to 75;the area on the right of the mean; it is the probability of having an average score between 70 and 75) We are interested on the total area ANS: .2794 + .1480 = .4274 42.74% of NYS high school students have average scores between 60 and 75. Percentiles: Bus310 - Homework expectedvalue-normaldistribution Here is a definition of percentile from Wikipedia: "In statistics, a percentile (or centile) is the value of a variable below which a certain percent of observations fall. For example, the 20th percentile is the value (or score) below which 20 percent of the observations may be found. " A percentile represents a value. If for example somebody's GPA score falls in the 93%ile (93 percentile) it means that 93% of the students who took the test received a lower score. It is important to acknowledge the fact that the mean represents the 50 th percentile. This will give you an idea in terms of where the other values for other specific percentiles fall. For example any percentile 50%, will be represented by a value greater than the mean and it will be (on the x axis) to the right of the mean value. The normalized value (z-score) will be a positive value. On the other hand any percentile