Chapter 5

1. Chapter 5 Sampling Distributions

2. Chapter 5.1 Sampling Distributions of sample mean X-bar

3. Review: Chapter 1.3, Normal distribution An important property of a density curve is that areas under the curve correspond to relative frequencies Example: The National Collegiate Athletic Association (NCAA) requires Division I athletes to score at least 820 on the combined math and verbal SAT exam to compete in their first college year. The SAT scores of 2003 were approximately normal with mean 1026 and standard deviation 209. What proportion of all students would be NCAA qualifiers (SAT ≥ 820)?

4. Sample: The part of the population we actually examine and for which we do have data. A statistic is a number describing a characteristic of a sample. Review Chap3: Population versus sample • Population: The entire group of individuals in which we are interested but can’t usually assess directly. • A parameter is a number describing a characteristic of the population. Population Sample

5. Objectives (chapter 5.1) Sampling distribution of a sample mean • Sampling distribution of sample mean (x-bar) • For normally distributed populations • The central limit theorem Question: In high school, when doing Physics and Chemistry experiments, why do we need to repeat an experiment for multiple times? Then take an average as our final experiment result. It sounds to only waist our time, energy and materials on the repetition. Is it correct?

6. Simple random sample (SRS) Data are summarized by statistics (mean, standard deviation, median, quartiles, correlation, etc..) Concerns: 1) Is sample mean related to population mean? 2) If yes, what will be the relationship? Or say, how far or how close is a sample mean away from the population mean?

7. Sampling Distribution of sample mean of 10 random digits • Select 10 random digits from Table B, and then take the sample mean; • Repeat this process 25 times for each student from Dr. Chen’s class. More details with illustration: 1. Based on Table B (random digit table), we randomly select a line, for example line 106 in this case: 2. Take sample average of random digits of (6, 8, 4, 1, 7, 3, 5, 0, 1, 3). We will have sample mean as sample mean #1=(6+8+4+1+7+3+5+0+1+3) /10=3.8; Now we move forward to another set of 10 random digits of (1, 5, 5, 2, 9, 7, 2, 7, 6, 5). We will have the sample mean as sample mean #2=(1+5+5+2+9+7+2+7+6+5) /10=4.9; Repeat this procedure 25 times until you get sample mean #25.

8. Sampling Distribution of sample mean of 10 random digits (2)2.2 2.2 (7)2.9 2.9 3.0 3.0 3.0 3.0 3.0 (8)3.1 3.1 3.2 3.2 3.3 3.4 3.4 3.5 (12)3.6 3.6 3.7 3.8 3.8 3.8 3.8 3.8 3.8 3.8 3.9 4.0 (16)4.1 4.1 4.2 4.2 4.2 4.3 4.3 4.3 4.3 4.4 4.4 4.4 4.5 4.5 4.5 4.5 (10)4.6 4.7 4.7 4.7 4.8 4.9 4.9 4.9 4.9 4.9 (8)5.2 5.2 5.3 5.3 5.3 5.3 5.5 5.5 (4)5.9 6.0 6.0 6.0 (4)6.2 6.2 6.3 6.4 (1)6.8 Q: Draw a histogram with classes as:

9. Sampling Distribution of sample mean of 10 random digits Sampling distribution of “x bar” Histogram of some sample averages Q: Write a journal about how to get the distribution of Sample mean X-bar today, to summarize in-class activity we have and ideas.

10. Sampling Distribution 1st Sample Sample mean 8 Select 10 random digits from Table B 4 6 3 3 = 6 8 4 9 8 9 7 2 0 8 5 4 2nd Sample 7 3 7 6 1 8 6 = 4.5 4 2 0 7 6 Population 2 3 25th Sample 1 8 9 5 6 1 3 4 = 4.6 0 9 There is some variability in values of a statistic over different samples.

11. Population Distribution for 10 random digits Population distribution of 0-9 random digits

12. Sampling Distribution of sample mean of 10 random digits • Select 10 random digits from Table B, and then take the sample mean; • Repeat this process 25 times for each students Spring 2012. • Make a histogram of sample mean’s from the class with 1098 X-bar’s. The probability distribution looks like a Normal distribution. The probability distribution of a statistic is called its sampling distribution. Sampling distribution of “x bar” Histogram of some sample averages For the histogram: Center of X-bar = 4.541 SD of X-bar = 0.9 Min. 1st Qu. Median Mean 3rd Qu. Max. 1.400 3.600 4.400 4.451 5.400 7.800

13. Sampling Distribution of sample mean of 10 random digits 8 4 6 3 3 8 4 9 8 9 7 2 2 8 7 0 6 2 5 4 1 3 8 0 5 4 7 3 7 8 6 1 6 4 2 0 7 6 Population 3 2 3 2 1 6 8 7 3 9 5 6 9 1 1 4 5 3 3 4 0 9 Center of X-bar = 4.5 SD of X-bar = 0.9

14. For any population with mean m and standard deviation s: • The mean, or center of the sampling distribution of x bar, is equal to the population mean m : • The standard deviation of the sampling distribution is s/√n, where n is the sample size : Mean and standard deviation of a sample mean Sample Mean’s are less variable than individual observations.

15. For normally distributed populations When a variable in a population is normally distributed, the sampling distribution of x bar for all possible samples of size n is also normally distributed. Sampling distribution If the population is N(m, s) then the sample means distribution is N(m, s/√n). Population

16. Sampling distribution of a sample mean=distribution of Population

17. Example: Soda Drink Let X denote the actual volume of soda in a randomly selected can. Suppose X~N(12oz, 0.4oz), 16 cans are to be selected. a) The average volume is normally distributed with mean____ and standard deviation___. b) Find the probability that the sample average is greater than 12.1 oz. If the population is N(m, s) then the sample means distribution is N(m, s/√n). Mean of x-bar = 12; SD of x-bar = 0.1; P(Z>1) = 1-0.8413 = 0.1587.

18. Exercise 5.21, page 310 • Diabetes during pregnancy. A patient is classified as having gestational diabetes if the glucose level is above 140 mg/dl one hour after a sugary drink. Patient Sheila’s glucose level follows a Normal distribution with m=125 mg/dl, s=10 mg/dl. • (a) If a single glucose measurement is made, what is the probability that Sheila is diagnosed as having gestational diabetes. • (b) If measurements are made instead on three separated days and the mean result is compared with criterion 140 mg/dl, what is the probability that Sheila is diagnosed as having gestational diabetes. If the population is N(m, s) then the sample means distribution is N(m, s/√n). (b) n=3: If x is the mean of three measurements, then x-bar has a N(125, 10/√3 ) or N(125 mg/dl, 5.7735 mg/dl) distribution, and P(x > 140) = P(Z >2.60) = 0.0047. (a) n=1: Let X be Sheila’s measured glucose level. (a) P(X > 140) = P(Z > 1.5) = 0.0668.

19. For Normal distributed populations If the population is N(m, s) then the sample means distribution is N(m, s/√n). Concern: What will happen when sample size gets bigger and bigger?

20. Review----Sampling Distribution of sample mean of 10 random digits 8 4 6 3 3 8 4 9 8 9 7 2 2 8 7 0 6 2 5 4 1 3 8 0 5 4 7 3 7 8 6 1 6 4 2 0 7 6 Population 3 2 3 2 1 6 8 7 3 9 5 6 9 1 1 4 5 3 3 4 0 9 Center of X-bar = 4.5 SD of X-bar = 0.9

21. Population with strongly skewed distribution Sampling distribution of for n = 2 observations Sampling distribution of for n = 10 observations Sampling distribution of for n = 25 observations Central Limit Theorem (CLT) m s

22. For Non-Normal distributed populations CLT says that: Even if the population is NOT Normal, but with mean m and SD s, when sample size is large enough, the sample means distribution is N(m, s/√n) approximately. Concern: What will happen when sample size gets bigger and bigger?

23. Sampling distribution of a sample mean=distribution of Population

24. IQ scores: population vs. sample In a large population of adults, the mean IQ is 112 with standard deviation 20. Suppose 200 adults are randomly selected for a market research campaign. • The distribution of the sample mean IQ is:  A) Exactly normal, mean 112, standard deviation 20  B) Approximately normal, mean 112, standard deviation 20  C) Approximately normal, mean 112 , standard deviation 1.414 D) Approximately normal, mean 112, standard deviation 0.1 C) Approximately normal, mean 112 , standard deviation 1.414  Population distribution : N(112; 20) Sampling distribution for n = 200 is N(112; 1.414)

25. Examples #5.12, page 309 • Songs on an iPod. An ipod has about 10,000 songs. The distribution of the play time for these songs is highly skewed. Assume that the standard deviation for the population is 280 seconds. • (a) What is the standard deviation of the average time when you take an SRS of 10 songs from this population? • (b) How many songs would you need to sample if you wanted the standard deviation of x-bar to be 15 seconds? (a) The standard deviation is σ/√10 = 280/√10 ~ 88.5438 seconds. (b) In order to have σ/√n = 280/√n = 15 seconds, we need √n = 280/15 ~ 18.667, so n ~ (18.667)^2 = 348.5 — use n = 349.

26. Example: children’s attitudes toward reading • In the journal Knowledge Quest (Jan/Feb 2002), education professors at the University of Southern California investigated children’s attitudes toward reading. One study measured third through sixth graders’ attitudes toward recreational reading on a 140-point scale. The mean score for this population of children was 106 with a standard deviation of 16.4. • In a random sample of 36 children from this population, • a) what is the sampling distribution of x-bar? • b) find P(x<100).

27. Answer to Example 4 • Z=-2.20 • Probability=normalcdf(-E99, -2.20, 0, 1)=0.0139

28. More Exercise on Chapter 5.1: 1. You were told that the weight of a new born baby follows normal distribution with mean 7 pounds and SD 0.5 pounds. The average weight of the next 16 new born in your local hospital is around ______, with SD _____. what’s the prob that the average is between 7.2 and 7.5 pounds? 2. The carbon monoxide in a certain brand of cigarette (in milligrams) follows normal distribution with mean 12 and SD 1.8. For 40 randomly selected cigarettes, a) What is the sampling distribution of sample mean? b) Find the prob that the average carbon monoxide is between 10 and 13. 3. The amount of time that a drive-through bank teller spends on a customer follows normal distribution with mean 4 minutes and SD 1.5 minutes. For the next 50 customers, find the prob that the average time spent is more than 5 minutes 4. The rate of water usage per hour (in Thousands of gallons) by a community follows normal distribution with mean 5 and SD 2. For the next 30 hours, • What is the sampling distribution of sample mean? • Find the probability that the average rate of usage per hour is less than 4? Answer: 1. new SD=0.125, Z7.2=1.6, Z7.5=4, area=1-0.9452=0.0548 2. new SD=0.285, Z10=-7.02, Z13=3.5, area is almost 100% 3. new SD=0.212, Z5=4.72, area is almost zero. 4. new SD=0.365, Z4=-2.74, area=1-0.9452=0.0031. EX: 5.7, 5.8, 5.18(a-c), 5.24, 5.21,5.12

29. Chapter 5.2 Sampling Distributions of sample proportion p-hat

30. Review: Sampling proportion p-hat Sample proportion: (p-hat, or relative frequency) Population proportion:

31. Reminder from Chapter 3: Sampling variability Each time we take a random sample from a population, we are likely to get a different set of individuals and calculate a different statistic. This is called sampling variability. If we take a lot of random samples of the same size from a given population, the variation from sample to sample—the sampling distribution—will follow a predictable pattern.

32. Sampling Distribution of sample proportion of 10 random digits • Select 10 random digits from Table B, and then take the sample proportion of EVEN numbers; • Repeat this process 4 times for each student from Dr. Chen’s class. More details with illustration: 1. Based on Table B (random digit table), we randomly select a line, for example line 106 in this case: 2. Take sample proportion of EVEN numbers of random digits of (6, 8, 4, 1, 7, 3, 5, 0, 1, 3). We will have sample proportion of EVEN #’s and gives sample proportion #1 = 4/10=0.4; Now we move forward to another set of 10 random digits of (1, 5, 5, 2, 9, 7, 2, 7, 6, 5), and we will have sample mean and gives sample proportion #2 = 3 /10=0.3; Repeat this procedure 4 times until you get sample proportion #4.

33. Sampling Distribution of sample mean of 10 random digits (1)0.1 (2)0.2 0.2 (5)0.3 0.3 0.3 0.3 0.3 (21)0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 (17)0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 (17)0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 (7)0.7 0.7 0.7 0.7 0.7 0.7 0.7 (5)0.8 0.8 0.8 0.8 0.8 (1)0.9 Q: Draw a histogram with classes as: (for line 101-120 in Table B)

34. Sampling Distribution of sample mean of 10 random digits Q: Write a journal about how to get the distribution of Sample proportion p-hat, to summarize in-class activity we have and ideas we get today. Sampling distribution of “p hat” Histogram of some sample proportion

35. Sampling Distribution 1st Sample Sample proportion 8 Select 10 random digits from Table B and find sample proportion of even # 4 6 3 3 8 4 9 8 9 7 2 0 8 5 4 2nd Sample 7 3 7 6 1 8 6 4 2 0 7 6 Population 2 3 25th Sample 1 8 9 5 6 1 3 4 0 9 There is some variability in values of a statistic over different samples.

36. Sampling Distribution of sample proportion of even # of 10 random digits • Select 10 random digits from Table B, and then take the sample proportion of even #. • Repeat this process a lot of times, say 10,000 times. • Make a histogram of these 10,000 sample mean’s. The probability distribution looks like a Normal distribution. The probability distribution of a statistic is called its sampling distribution. Sampling distribution of “p-hat” Histogram of some sample proportion Center of p-hat = 0.5018 SD of p-hat = 0.1598 Note: n=10. SD of p-hat =

37. Sampling distribution of the sample proportion The sampling distribution of is never exactly normal. But as the sample size increases, the sampling distribution of becomes approximately normal. The normal approximation is most accurate for any fixed n when p is close to 0.5, and least accurate when p is near 0 or near 1.

38. Sampling Distribution of • If data are obtained from a SRS and np>10 and n(1-p)>10, then the sampling distribution of has the following form: • For sample percentage: • is approximately normal with mean p and standard deviation:

39. Sampling distribution of a sample Proportion=distribution of

40. The Gallup Organization surveyed 1,252 debit cardholders in the U.S. and found that 180 had used the debit card to purchase a product or service on the Internet (Card Fax, November 12, 1999). Suppose the true percent of debit cardholders in the U.S. that have used their debit cards to purchase a product or service on the Internet is 15%. • Calculate p hat (sample proportion ). • The sample proportion (p hat ) is approximately normal with mean = ______ and standard deviation = ______. • Find the probability of getting a sample proportion smaller than 14.4%. ANS: Z=(0.144-0.15)/0.01=-0.6 Pr(Z<-0.6)= normalcdf(-E99, -0.6, 0, 1) = 0.2743

41. Example 2 Maureen Webster, who is running for mayor in a large city, claims that she is favored by 53% of all eligible voters of that city. Assume that this claim is true. In a random sample of 400 registered voters taken from this city: a.) What is the sampling distribution of p-hat? Population proportion p= _________. The sample proportion is approximately normal with mean = ______ and standard deviation = ______. b) What is the probability of getting a sample proportion less than 49% in which will favor Maureen Webster? c.) Find the probability of getting a sample proportion in between 50% and 55%. (c) Z=(0.5-0.53)/0.02495 = -1.20; Z=(0.55-0.53)/0.02495 = 0.80; Pr(-1.20 <Z<0.80) =normalcdf(-1.20, 0.80, 0, 1) =0.673 (b) Z=(0.49-0.53)/0.02495 = -1.60 Pr(Z<-1.60) =normalcdf(-E99, 01.6, 0, 1) = 0.0548

42. More Exercise on Chapter 5.2: 1. 30% of all autos undergoing an emissions inspection at a city fail in the inspection. Among 200 cars randomly selected in the city, the percentage of cars that fail in the inspection is around_____, with SD______. Find the prob that the percentage is between 31% and 35%. 2. 60% of all residents in a big city are Democrats. Among 400 residents randomly selected in the city, a) What is the sampling distribution of p-hat? b) Find Pr(sample percentage<58%) 3. In airport luggage screening it is known that 3% of people have questionable objects in their luggage. For the next 1600 people, use normal approximation to find the prob that at least 4% of the people have questionable objects. 4. It is known that 60% of mice inoculated with a serum are protected from a certain disease. If 80 mice are inoculated, a) What is the sampling distribution of p-hat? b) find the prob that at least 70% are protected from the disease. HWQ: 5.22, 5.23(a,b) 5.73

43. Sampling distribution of a sample mean=distribution of Population

44. Summary to Chapter 5 1. If X~N(µ, σ) exactly, then a) what is the mean of X-bar? b) what is SD of X-bar? c) what is the sampling distribution of X-bar? 2. If X is NOT normal, but with population mean µ and population SD σ. When sample size is big enough, a) what is the mean of X-bar? b) what is SD of X-bar? c) what is the sampling distribution of X-bar? 3. With population proportion p and sample size n, a) what is the mean of p-hat? b) what is SD of p-hat? c) what is the sampling distribution of p-hat?