Lecture 11 Lecture Outline 1. Collect Homework 2. Z-score Look Up Quiz 3. Sampling Distribution Activity 4. Sample, Population and Sampling Distribution Activity a. Empirical vs. Theoretical b. Know vs. Unknown 5. Central Limit Theorem Assignments 1. Sampling Distribution Activity 2. Sampling Distribution Concepts 3. Sampling Distribution Problems Preparation for Next Class 1. Read Chapter 7 Quiz for Next Class- Distributions (definition, notation and properties)
Sampling Distribution Class Activity This is my population: x Frequency 0 1 1 2 2 4 3 5 4 5 5 4 6 2 7 1 Total 24 The population mean and standard deviation are: = = 1.71 These are the 3 samples I took during class (N=10): Sample 1 Sample 2 Sample 3 These are the sample means of each of my samples: = = = How do the sample means compare to the population mean? This is the mean and standard deviation of the sampling distribution: = = How does the sampling distribution mean compare to the population mean? How does the standard error compare to the population mean?
Sampling Distribution Concepts For 1-6, use the Central Limit Theorem to find the standard error given the standard deviation and sample size. 1) =10, =100 standard error = 2) =200, =92 standard error = 3) =9, =1003 standard error = 4) =26, =53 standard error = 5) =3.4, =61 standard error = 6) =9.75, =43 standard error = For 7-10 answer in full sentences and in your own words. 7) What is a sampling distribution? 8) What does the Central Limit theorem tell us about a sampling distribution? 9) What is the standard error? 10) Define the population, sample and sampling distributions. What symbols are used for the mean and standard deviation of each distribution?
Sampling Distribution Problems 1) The Nome Ice Company was in business for only 3 days. Here are the numbers of telephone calls received on each of those days: 10, 6, 5. Assume that samples of size 2 are randomly selected with replacement from this population of 3 values. a. List the 9 different possible samples and find the mean of each of them. The list of means is the sampling distribution. b. Find the mean and standard deviation of the sampling distribution. c. Is the mean of the sampling distribution equal to the mean of the population? 2) Here are the number of sales made per day by Kim Ryan, a courteous telemarketer who worked for 4 days before being fired: 1, 11, 9, 3. Assume that samples of size 2 are randomly selected with replacement from this population of 4 values. a. List the 16 different possible samples and find the mean of each of them. The list of means is the sampling distribution. b. Find the mean and standard deviation of the sampling distribution. c. Is the mean of the sampling distribution equal to the mean of the population?
3) Here is the population of all five US Presidents who had professions in the military, along with their ages at inauguration: Eisenhower (62), Grant (46), Harrison (68), Taylor (64) and Washington (57). Assume that samples of size 2 are randomly selected with replacement from this population of 5 ages. a. List the 25 different possible samples and find the mean of each of them. The list of means is the sampling distribution. b. Find the mean and standard deviation of the sampling distribution. c. Is the mean of the sampling distribution equal to the mean of the population?
4) Assume women s heights are normally distributed with a mean of 63.6 inches and a standard deviation of 2.5 inches. a. If 1 woman is randomly selected find the probability that her height is less than 64 inches. b. If 36 women are selected, find the probability that they have a mean height less than 64 inches. c. If 1 woman is randomly selected find the probability that her height is greater than 63 inches. d. If 100 women are selected, find the probability that they have a mean height greater than 63 inches. e. If 1 woman is randomly selected find the probability that her height is between 63.5 and 64.5 inches. f. If 9 women are selected, find the probability that they have a mean height between 63.5 and 64.5 inches. g. If 1 woman is randomly selected find the probability that her height is between 60 and 65 inches.
h. If 16 women are selected, find the probability that they have a mean height between 60 and 65 inches. 5) The new lucky casino wants to increase revenue by providing buses that can transport gamblers from other cities. Research shows that these gamblers tend to be older, they tend to play slot machines only, and they have losses with a mean of $182 and a standard deviation of $105. The buses carry 35 gamblers per trip. The casino gives each bus passenger $50 worth of vouchers that can be converted to cash, so the casino needs to recover that cost in order to make a profit. Find the probability that if the bus is filled with 35 passengers, the mean amount lost by a passenger with exceed $50. Based on this result, does the casino gamble when it provides these buses?