AP Stats ~ Lesson 6B: Transforming and Combining Random variables OBJECTIVES: DESCRIBE the effects of transforming a random variable by adding or subtracting a constant and multiplying or dividing by a constant. FIND the mean and standard deviation of the sum or difference of independent random variables. FIND probabilities involving the sum or difference of independent Normal random variables.
In Section 6.1, we learned that the mean and standard deviation give us important information about a random variable. In this section, we ll learn how the mean and standard deviation are affected by transformations on random variables. In Chapter 2, we studied the effects of linear transformations on the shape, center, and spread of a distribution of data. Recall: 1. Adding (or subtracting) a constant, a, to each observation: Adds a to measures of center and location. Does not change the shape or measures of spread. 2. Multiplying (or dividing) each observation by a constant, b: Multiplies (divides) measures of center and location by b. Multiplies (divides) measures of spread by b. Does not change the shape of the distribution.
Example: El Dorado Community College considers a student to be full-time if he or she is taking between 12 and 18 units. The number of units X that a randomly selected El Dorado Community College full-time student is taking in the fall semester has the following distribution. Here is a histogram of the probability distribution along with the mean and standard deviation: At El Dorado Community College, the tuition for full-time students is $50 per unit. That is, if T = tuition charge for a randomly selected full-time student, T = 50X. Here is the probability distribution for T and a histogram of the probability distribution: The mean and standard deviation of this distribution are 600(0.25) 650(0.10) (900)(0.15) $732.50 T T 10,568 $103 What do you notice about the 2 distributions?
As with regular data, if we multiply a random variable by a negative constant b, our common measures of spread are multiplied by b.
Example: In addition to tuition charges, each full-time student at El Dorado Community College is assessed student fees of $100 per semester. If C = overall cost for a randomly selected full-time student, C = 100 + T. Here is the probability distribution for C and a histogram of the probability distribution: The mean and standard deviation of this distribution are What do you notice when you compare this to the distribution that was adjusted for tuition?
Check your understanding: A large auto dealership keeps track of sales made during each hour of the day. Let X= the number of cars sold during the first hour of business on a randomly selected Friday. Based on previous records, the probability distribution of X is as follows: Cars Sold: 0 1 2 3 Probability: 0.3 0.4 0.2 0.1 The random variable X has the mean 1.1 and standard deviation 0.943. A) Suppose that the dealerships manager receives a $500 bonus from the company for each car sold. Let Y=the bonus received from car sales during the first hour on a randomly selected Friday. Find the mean and standard deviation of Y. B) To encourage customers to buy cars on Friday mornings, the manager spends $75 to provide coffee and doughnuts. The manager s net profit T on a randomly selected Friday is the bonus earned minus this $75 Find the mean and standard deviation of T.
Linear transformations have similar effects on other measures of center or location (median, quartiles, percentiles) and spread (range, IQR). Whether we re dealing with data or random variables, the effects of a linear transformation are the same. Note: These results apply to both discrete and continuous random variables.
Example: In a large introductory stats class, the distribution of raw scores on a test X follows a Normal distribution with a mean of 17.2 and a standard deviation of 3.8. The professor decides to scale the scores by multiplying the raw scores by 4 and adding 10. a) Define the variable Y to be the scaled score of a randomly selected student from this class. Find the mean and standard deviation of Y. b) What is the probability that a randomly selected student has a scaled test score of at least 90?
Many interesting statistics problems require us to examine two or more random variables and their relationship. Let s revisit El Dorado Community College. They also have a campus downtown, specializing in just a few fields of study. Full-time students at the downtown campus take only 3-unit classes. Let Y=number of units taken in the fall semester by a randomly selected full-time student at the downtown campus. Here is a probability distribution of Y and a probability histogram. The mean of this distribution in 15 units, and the standard deviation is 2.3 units. If you were to randomly select one full-time student from the main campus and one full-time student from the downtown campus and add their number of units, the expected value of the sum (S=X+Y) would be the mean of X plus the mean of Y = 14.65 + 15 = 29.65
The only way to determine the probability for any value of T is if X and Y are independent random variables. If knowing whether any event involving X alone has occurred tells us nothing about the occurrence of any event involving Y alone, and vice versa, then X and Y are independent random variables. Probability models often assume independence when the random variables describe outcomes that appear unrelated to each other. You should always ask whether the assumption of independence seems reasonable. In our Community College example, it is reasonable to assume X and Y are independent since the campuses are in different parts of the town.
When we add two independent random variables, we find that their variances add. Standard deviations do not add. Remember that you can add variances only if the two random variables are independent, and that you can NEVER add standard deviations!
We can perform a similar investigation to determine what happens when we define a random variable as the difference of two random variables. In summary, we find the following:
Example: Let B=the amount spent on books in the fall semester for a randomly selected full-time student at El Dorado Community College. Suppose that the mean amount for B is 153 and the standard deviation of B is 32. Recall from earlier that C=overall cost of tuition and fees for a randomly selected full-time student at El Dorado Community College and that the mean amount was $832.50 and the standard deviation is $103. Find the mean and standard deviation of the cost of tuition, fees, and books for a randomly selected full-time student.
Example: Let s return to the auto dealership. The dealership keeps track of sales and lease agreements made during each hour of the day. Let X = number of cars sold and Y = number of cars leased during the first hour of business on a randomly selected Friday. The mean of X is 1.1, and its SD is 0.943. The mean of Y is 0.7, and its SD is 0.64. Define D = X Y. Assume that X and Y are independent. A) Find and interpret the mean of D. B) Find and interpret the standard deviation of D. Show your work. C) The dealership s manager receives a $500 bonus for each car sold and a $300 bonus for each car leased. Find the mean and standard deviation of the difference in the manager s bonus for cars sold and leased.
If a random variable is Normally distributed, we can use its mean and standard deviation to compute probabilities. Any sum or difference of independent Normal random variables is also Normally distributed. What does this mean? We can use the process of normalizing the data using z-scores, and then find the area beneath the curve to determine probability.
Example: Suppose that a certain variety of apples have weights that follow a normal distribution with a mean of 9 ounces and a standard deviation of 1.5 ounces. If bags of apples are filled by randomly selecting 12 apples, what is the probability that the sum of the weights of the 12 apples is less than 100 ounces? Step 1: Find the new mean and standard deviation of the BAG of apples. Step 2: Using the new mean and SD, perform the calculations and find the probability that the sum of the weight of the apples is less than 100 ounces. (Don t forget to answer in the context of the problem.)
Example: To save time and money, many single people have decided to try speed dating. At a speed dating event, women sit in a circle and men spend about 10 minutes getting to know a woman before moving on to the next one. Suppose that the height M of male speed daters follows a Normal distribution with a mean of 69.5 inches and an SD of 4 inches, and that the height F of female speed daters follows a Normal distribution with a mean of 65 inches and an SD of 3 inches. What is the probability that a randomly selected male speed dater is taller than the randomly selected female speed dater with whom he is paired?
Homework: Page 382: #35-63 odds, 65, 66 Read pp. 386-410