Chapter 3. What Do Samples Tell Us? Chapter 3 1

Transcription

1 Chapter 3 What Do Samples Tell Us? Chapter 3 1

2 Thought Question 1 During a medical exam, the doctor measures your cholesterol two times. Do you think both measurements would be exactly the same? Why or why not? Chapter 3 2

3 Thought Question 2 To estimate the percentage of all adults who have an Internet connection in their homes, a properly chosen sample of 1100 adults across the U.S. was sampled, and 60% said yes. How close do you think that is to the percentage of the entire country who have an Internet connection? Within 30%? 10%? 5%? 1%? Exactly the same? Chapter 3 3

4 Sampling Terminology Parameter fixed, unknown number that describes the population A parameter is a fixed number, but in practice we don t know its value Statistic known value calculated from a sample It can change from sample to sample a statistic is used to estimate a parameter Bias in repeated samples, the sample statistic consistently misses the population parameter in the same direction Variability different samples from the same population may yield different values of the sample statistic Chapter 3 4

5 Figure 3.3 Bias and variability in shooting arrows at a target. Bias means the archer systematically misses in the same direction. Variability means that the arrows are scattered.

6 Sampling Strategy A good sampling method has both small bias and small variability. To reduce bias, use random sampling. To reduce variability, use larger samples estimates from random samples will be closer to the true values in the population if the samples are larger how close will they be? margin of error Chapter 3 6

7 Proportions The proportion of a population that has some outcome ( success ) is p. The proportion of successes in a sample is measured by the sample proportion: pˆ number of successes in the total number of observatio ns in sample the sample p-hat Chapter 3 7

8 Example: Same-sex marriages In 2004 President Bush made the declaration: Today I call upon the Congress to promptly pass, and send to the states for ratification, an amendment to our Constitution defining and protecting marriage as a union of a man and woman as husband and wife. A Gallup Poll conducted from July 2003 to February 2004 asked the following question: Would you favor or oppose a constitutional amendment that would define marriage as being between a man and a woman, thus barring marriages between gay and lesbian couples? Chapter 3 8

9 Example: Same-sex marriages Gallup Poll found out that such an amendment is supported by a very slim majority of Americans, 51%, with 45% opposed. Gallup talked with 2527 randomly selected adults to reach these conclusions. How can 2527 people tell us about the opinions of the 307 million people? A sample won t tell us the truth about the population, so Gallup gives a margin of error. Here is Gallup s claim: For results based on a sample of this size, one can say with 95% confidence that the error attributable to sampling and other random effects could be plus or minus 2 percentage points. Chapter 3 9

10 Margin of Error The amount by which the proportion obtained from the sample ( pˆ ) will differ from the true population proportion (p) rarely exceeds the margin of error. Chapter 3 10

11 What Margin of Error means Margin of Error plus or minus two percentage points is shorthand for the statement: If we took many samples using the same method we used to get this one sample, 95% of the samples would give a result within plus or minus 2 percentage points of the truth about the population. Chapter 3 11

12 Chapter 3 12

13 Margin of Error Chapter 3 13

14 Margin of Error and Sample Size Chapter 3 14

15 Figure 3.1 The results of many SRSs have a regular pattern. Here, we draw 1000 SRSs of size 100 from the same population. The population proportion is p = 0.5. The sample proportions vary from sample to sample, but their values center at the truth about the population.

16 Figure 3.2 Draw 1000 SRSs of size 2527 from the same population as in Figure 3.1. The 1000 values of the sample proportion are much less spread out than was the case for smaller samples.

17 Confidence statements A confidence statement has two parts: a margin of error and a level of confidence. The margin of error says how close the sample statistic lies to the population parameter. The level of confidence says what percentage of all possible samples satisfy the margin of error. Chapter 3 17

18 Population size doesn t matter The variability of a statistic from a random sample does not depend on the size of the population as long as the population is at least 100 times larger than the sample. Chapter 3 18

19 Case Study Guaranteed Health Insurance in the U.S.? New York Times/CBS News Poll, January % say it should be guaranteed by the government same as in 2000, up 6 points from % say it is not the responsibility of the government Chapter 3 19

20 Case Study How the Poll was Conducted This New York Times/CBS News poll was based on telephone interviews conducted January 20 through January 25, 2006 with 1,229 adults throughout the United States. The survey has a random sampling error of approximately ±3 percent. Chapter 3 20

21 Case Study Conclusion (Confidence statement) For the proportion of the population who favor guaranteed health insurance, the sample proportion was pˆ =.62 (62%) and the margin of error was ±.03 (3%). We can then say that we are 95% confident that the proportion of the population who favor guaranteed health insurance was between.59 and.65 (59% and 65%). Chapter 3 21

22 Key Concepts Parameter versus Statistic Bias and Variability Margin of Error Confidence Statements Chapter 3 22

23 Exercise 3.13 Which data have high/low bias, high/low variability? Chapter 3 23

24 Exercises Determine whether the highlighted number is a parameter or a statistic. The Bureau of Labor Statistics announces that last month it interviewed all members of the labor force in a sample of 55,000 households; 4.7% of the people were unemployed. A carload lot of ball bearings has an average diameter of centimeters (cm). This is within the specifications for acceptance of the lot by the purchaser. The inspector happens to inspect 100 bearings from the lot with an average diameter of cm. This is outside the specified limits, so the lot is mistakenly rejected. Voter registration records show that 15.5% of all voters in Philadelphia are registered as Republicans. However, a radio talk show host in Philadelphia found that of 20 local residents who called the show recently, 60% were registered Republicans. A national polling organization uses a random digit dialing device to dial residential phone numbers in the U.S. Of the first 100 numbers dialed, 43 are unlisted numbers. This is not surprising, because 40% of all residential phones in the U.S. are unlisted. Chapter 3 24

25 Exercise 3.9 Canada s national health care. The Ministry of Health in the Canadian province of Ontario wants to know whether the national health care system is achieving its goals in the province. Much information about health care comes from patient records, but that resource doesn t allow us to compare people who use health services with those who don t. So the ministry of Health conducted the Ontario Health Survey, which interviewed a random sample of 61,239 people who live in the province of Ontario. a) What is the population for this sample survey? What is the sample? b) The survey found that 76% of males and 86% of females in the sample had visited a general practitioner at least once in the past year. Do you think these estimates are close to the truth about the entire population? Why? Chapter 3 25

26 Exercise 3.11 Sampling variability. In thinking about Gallup s sample of size 2527, we asked Could it happen that one random sample finds that 51% of adults favor an amendment and a second random sample finds that only 37% favor one?. Look at the following figure which shows the results of 1000 samples of this size when the population truth is p=0.5 or 50%. Would you be surprised if a sample from this population gave 51%? Would you be surprised if a sample gave 37%? Chapter 3 26

27 Exercise 3.14 Is a larger sample size always better? In February 2004, USA Today conducted an online poll. Visitors to their website were asked the following question: Should the U.S. pass a constitutional amendment banning gay marriage? Visitors could vote by clicking a button. The results as of 3:30 PM on February 25 were that 68.61% voted No and 31.39% voted Yes A total of 63,046 votes had been recorded. Using our quick method, we find that the margin of error for 95% confidence for a sample of this size is roughly equal to four-tenths of one percentage points. Is it correct to say that, based on this USA Today online poll, we are 95% confident that plus or minus 0.4% of American adults are opposed to having the U.S. pass a constitutional amendment banning gay marriage? Chapter 3 27

28 Exercise 3.22 The death penalty. The Gallup Poll asked a sample of 1010 adults, Are you in favor of the death penalty for a person convicted of murder? The proportion who said in favor was 69%. a) How many of the 1010 people interviewed said they were in favor of the death penalty for person convicted of murder? b) Gallup says that the margin of error for this poll is plus or minus 3 percentage points. Explain to someone who knows no statistics what margin of error plus or minus 3 percentage points means. Chapter 3 28

29 Exercise 3.24, and 3.28 Find the margin of error. Exercise 3.22 concerns a Gallup Poll sample of 1010 people. a)use the quick method to estimate the margin of error for statements about the population of all adults. b)is your result close to the 3% margin of error announced by Harris? c)suppose you want a margin of error half as large as the one you found in (a). How many people must you interview? Chapter 3 29

30 Exercise 3.29 Satisfying the Congress. Exercise 3.12 describes a sample survey of 1004 adults, with margin of error plus or minus 4 percentage points for 95% confidence. a) A member of the Congress thinks that 95% confidence is not enough. He wants to be 99% confident. How would the margin of error for 99% confidence based on the sample compare with the margin of error for 95% confidence. b)another member of the Congress is satisfied with 95% confidence, but she wants a smaller margin of error than plus or minus 4 percentage points. How can we get a smaller margin of error, still with 95% confidence? Chapter 3 30