Math 3339 Section 19377 MWF 10-11am SR 116 (and section 20582 – online) Bekki George [email protected] 639 PGH Office Hours: 11:00 - 11:45am MWF or by appointment EMCF 15 1.If a sample has a mean of 48 and a standard deviation of 3.2, what is the value in the set that corresponds to a z-score of -1.2? a. 51.84 b.53.12 c. 46.40 d.44.16 e. none of these 2. Find P(−1.9 < Z < 1.2) a. 0.8020 b.0.7659 c. 0.8562 d.0.9713 e. none of these 3. Find c such that P(Z > c) = 0.8790 a. -0.19 b.1.17 c. -1.17 d.0.19 e. none of these Section 5.3 - Statistics and Their Distributions Def: A statistic is any quantity whose value can be calculated from sample data. Prior to obtaining the data, there is uncertainty as to what value of any particular statistic will result. Therefore, a statistic is a random variable and will be denoted by an uppercase letter. A lowercase letter represents the calculated or observed value of the statistic. Def: The rv’s X1 , X 2 , X 3 ,..., X n are said to form a (simple) random sample of size n if 1.The X i 's are independent random variables. 2.Every X i has the same probability distribution. Ex: A certain brand of MP3 player comes in three configurations: 2GB ($80), 4GB ($100), and 8GB ($120) Let X = the cost of a single randomly selected purchase of the MP3 player. Suppose X has pmf given by the table below: x p(x) 80 0.2 100 0.3 120 0.5 Suppose, on a particular day, 2 of these MP3 players are sold. Let X 1 = selling price of first MP3 player X 2 = selling price of second MP3 player Determine the possible values for X = X1 + X 2 2 X1 + X 2 Give the pmf of X = 2 Compute the expected value and variance of X1 + X 2 X= 2 Section 5.4 - The Distribution of the Sample Mean Suppose we take n measurements from a normal distribution with unknown mean. How can we approximate the mean? Ex: Suppose the scores on Exam 1 are normally distributed and the average score is unknown. If 5 of the scores are 74 98 83 70 91 approximate the average score on the exam. What about the spread? Will it be the same as the population of exams? A simple example: We will be looking at the “sampling distribution of the means” for a set of data – this is the same as the “distribution of the sample means”. Consider the population consisting of the values 3,5,9,11 and 14. µ = ________________ σ = ______________ Let’s take samples of size 2 from this population. The set above is the sampling distribution of size 2 for this population. Let’s take samples of size 3 from this population. The set above is the sampling distribution of size 3 for this population. List all the possible pairs/triples from 3,5,9,11 and 14 and find their means. pairs x triples µ x = ____, σ x = _____ x µ x = ____, σ x = _____ Compare µ x (the mean of the sample means) to µ. What do you notice about σ x ? Theorem: If we sample from a normal distribution with mean µ and standard deviation σ, then the distribution of X is ( ) N mean = µ ,standard deviation = σ / n . Ex: Suppose the scores on Exam 1 are normally distributed with mean 74, and standard deviation 10. If 5 scores are taken at random, determine the probability that the average of those 5 scores is less than 70. Ex: Suppose that the race times of a certain sprinter are normally distributed with mean 55 seconds and standard deviation 4 seconds. If a certain qualification round require the average of three sprints to be less than 54 seconds in order to advance, determine the probability that the sprinter advances. The Central Limit Theorem If X is the mean of a random sample X1 , X 2 ,..., X n from a distribution with 2 σ mean µ and finite variance , then the distribution of X −µ Z= σ/ n approaches the standard normal distribution as n approaches infinity. Current research indicates that the distribution of the life expectancies of a certain protozoan is normal with a mean of 48 days and a standard deviation of 10.2 days. Find the probability that a simple random sample of 49 protozoa will have a mean life expectancy of 49 or more days. Suppose that a random sample of size 64 is to be selected from a population with mean 42 and standard deviation 9. What is the approximate probability that x will be within 0.5 of the population mean? What is the approximate probability that x will be more than 0.5 away from the population mean? EMCF 15 A waiter estimates that his average tip per table is $20 with a standard deviation of $4. If we take samples of 9 tables at a time, calculate the following probabilities when the tip per table is normally distributed. 4. What is the probability that his average tip is more than $19? a. 0.599 b. 0.773 c. 0.707 d. none of these 5. What is the probability that his average tip is between $19 and $22? a. 0.599 b. 0.773 c. 0.707 d. none of these