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Statistics in Surveys and Polls Part 3: Point Estimation and Confidence Intervals

Chapter 3: The Problem with Samples (Part 2)

“Statistics almost always give you the wrong answer”

Sample ≠ Population

True even with good samples

Statistics ≠ Parameters

Statistics have bias and variability

To reduce bias, use (SRS) and proper statistics

To reduce variability, use a larger sample (n)

Population and Sample: Opinion about Military Action against ISIS

Population = Opinions of all Americans concerning military action

p = true proportion of all Americans in favor

F N N F F F N N N N F F F F F N F N F F F

N N F F F N N N N F F F F F N F N F

F N N F F F

F N N F F F N N F F

F N N F F F

F N N F

F N N F F F

Sample 1: Opinions of 100 Americans polled

Sample 2: Opinions of 100 Americans polled

Practice Showing that Statistics ≠ Parameters

Flip a coin 100 times, count the number of heads. (A head represents “in favor” of military action against ISIS.)

Proportion “in favor” observed from 100 responses.

0.5	0.53	0.55	0.5	0.52
0.51	0.49	0.6	0.53	0.43
0.55	0.51	0.5	0.47	0.45
0.49	0.49	0.45	0.48	0.5
0.58	0.52	0.47	0.43	0.47
0.52	0.51	0.58	0.52	0.61
0.51	0.49	0.42	0.52	0.51
0.56	0.49	0.51	0.59	0.38
0.49	0.53	0.45	0.54	0.44
0.44	0.5	0.46	0.55	0.54

Point estimates:

Single guess as to true value of parameter

Let p = 0.5.

Notice that only a few times correct and point estimates vary.

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Second level

Third level

Fourth level

Fifth level

Point estimation

The proportion of “in favors” you obtained from 100 “responses” represents the point estimate.

It is a single guess as to the true value of the parameter.

The point estimate is the value of the statistic.

The point estimator is the statistic which is used to estimate the parameter of interest.

What is the point estimator?

= sample proportion of 100 Americans polled in favor

What is the parameter of interest?

p = true population proportion of all Americans in favor

Types of errors

Sampling error (margin of error=MOE)

Error resulting from surveying some and not all of units in population

Coverage error (undercoverage)

Error resulting from not allowing all units in population to have equal chance of being selected

Measurement error (response error)

Error resulting from poor question wording or questions presented in a manner that inaccurate or uninterpretable answers are obtained

Nonresponse error

Error resulting when units who do not respond are different from units that do respond

Always have since took a sample

Telephone surveys (land lines) Internet polls (low SES)

Question issues; Lie

Not at home; Refuse

Solving the problem

Margin of Error (MOE) (Sampling error)– use to “expand” our statistic;

Statistic ± MOE

Point estimate + MOE

Statistic ± MOE has a better chance of hitting the parameter than the statistic alone; this chance is called confidence level

Statistic ± MOE is sometimes called a confidence interval or an interval estimate

The purpose of a confidence interval (CI) is to estimate true value of a parameter.

(Point estimate – MOE, Point estimate + MOE)

Point estimate is always in the confidence interval. It is in the center of the interval.

MOE

Called a 95% confidence interval

Solving the Problem (cont)

Calculating MOE

Qualitative

Estimate proportion (p) - 95% MOE ≈

Quantitative

Estimate mean (µ) - 95% MOE ≈

Reduce MOE by

increasing n

Use later

Interpreting Point estimate + MOE

Conclusion applies to population

Never certain

Doesn’t have to be 95%

http ://digitalfirst.bfwpub.com/stats_applet/stats_applet_4_ci.html

MOE

Use your point estimate to create a 95% confidence interval for p (the true proportion of Americans in favor of military action against ISIS).

Did your interval enclose p=0.5?

Point estimate + MOE

Suppose your point estimate was 0.58.

0.58 + 0.1 => (0.58-0.1, 0.58+0.1)

(0.48, 0.68)

This interval does enclose p=0.5.

= 0.1

Suppose your point estimate was 0.61.

0.61 + 0.1 => (0.61-0.1, 0.61+0.1)

(0.51, 0.71)

This interval does not enclose p=0.5.

These values are called point estimates.

2016 Presidential Poll Results

How was the MOE for the Detroit Free Press poll determined?

= 0.0408

Candidate	Trafalgar Group	Fox 2 Detroit/ Mitchell	Detroit Free Press
Donald Trump	0.49 + 0.028 (0.462, 0.518)	0.41 + 0.031 (0.379, 0.441)	0.38 + 0.04 (0.34, 0.42)
Hillary Clinton	0.47 + 0.028 (0.442, 0.498)	0.46 + 0.031 (0.429, 0.491)	0.42 + 0.04 (0.38, 0.46)

Give 95% confidence intervals for estimating the proportion of Michigan voters to vote for each candidate.

In the above table, circle the confidence interval(s) that did not include the actual election result. What do you notice?

Michigan 2016 Presidential Election Results

Donald Trump (R) 47.6%

Hillary Clinton (D) 47.3%

Point estimate

MOE

Largest MOE so smallest sample size, n

2016 Iowa Primary Polls

2016 Iowa Primary Election Results

Finding sample size

Sample size can be determined for a particular MOE

Suppose we can live with an MOE of 1% (+ .01)

If MOE = 0.01=

What is the sample size for an MOE of 3%?

Qualitative observations

What do you notice about the relationship between n and MOE?

n = = 1111

As n increases, the MOE decreases.

Finding sample size

Quantitative observations

Sample size

Population Size	Sample Size
100	92
200	169
400	291
600	384
800	458
1,000	517
2,000	696
4,000	843
6,000	906
8,000	942
10,000	965
20,000	1,013
40,000	1,040
100,000	1,056
1,000,000	1,066
1,000,000,000	1,067

Sample sizes determined based on population size, 3% margin of error and 95% confidence.

Dillman, D.A. (2007) Mail and Internet Surveys pg 207

Interpreting Political Polls

Results from a recent poll of the percentage of likely to vote registered voters who would vote for a particular candidate. Do you think A or B will win?

Candidate A 59%

Candidate B 41%

Sampling error of 3%

Candidate A: 0.59 + 0.03

Candidate B: 0.41 + 0.03

(0.56, 0.62)

(0.38, 0.44)

CI’s do not overlap.

Candidate A

Candidate B

0.59

0.41

0.62

0.56

0.38

0.44

Interpreting Political Polls

24%

12%

Predict Candidate A is likely to win by as much as 24% or as little as 12%

Interpreting Political Polls

Results from a recent poll of the percentage of likely to vote registered voters who would vote for a particular candidate. Do you think A or B will win?

Candidate A 45%

Candidate B 41%

Sampling error of 3%

Candidate A: 0.45 + 0.03

Candidate B: 0.41 + 0.03

(0.42, 0.48)

(0.38, 0.44)

CI’s do overlap.

Interpreting Political Polls

Candidate A

Candidate B

0.45

0.41

0.48

0.42

0.38

0.44

Predict there could be as much as a 10 point lead for Candidate A over Candidate B or a 2 point lead for Candidate B over Candidate A

“Statistical dead heat”, “within the margin of error” or “too close to call”

Popularity of Congress

From January 3-6, 2013, 803 Americans surveyed by telephone were asked:

“What do you have a higher opinion of:

Congress or _____?” MOE was 3.4%.

Congress	John Edwards	Congress CI	Edwards CI	Statistical difference?
45%	29%
Congress	Cockroaches	Congress CI	Cockroaches CI	Statistical difference?
43%	45%

http://www.publicpolicypolling.com/main/2013/01/congress-somewhere-below-cockroaches-traffic-jams-and-nickleback-in-americans-esteem.html

MOE was 3.4%

45% + 3.4%

(41.6%, 48.4%)

29% + 3.4%

(25.6%, 32.4%)

43% + 3.4%

(39.6%, 46.4%)

45% + 3.4%

(41.6%, 48.4%)

There is a real difference.

(CI’s do not overlap)

Can’t say there is a real difference. (CI’s overlap)

The Problem with Polls

Poll results must be interpreted with extra care. A great example of some of the problems can be found in this example, and this was a “good” poll. I have added to the example to make some points.

The Pew Research Center for the People and the Press imitated the methods of the better opinion polls. They dialed telephone numbers at random to get a “SRS” of 1000 household opinions on school choice. Here are the actual results

Numbers eliminated: business, FAX, no service,… not given

Households without phones unknown

Never answered phone 938 (33%)

Answered but refused to participate 678 (23%)

Not eligible: no person age 18, language barrier, … 221 ( 8%)

Incomplete interview 42 ( 1%)

Complete interview: in favor 451 (16%)

Complete interview: against 392 (14%)

Complete interview: don’t know, don’t care 157 ( 5%)

Total called 2879

Result was that 45% favored school choice vs. 39% against (with a sampling error of ± 2%).

Problems/Issues?

0.45+0.02 => (0.43, 0.47) 0.39+0.02 => (0.37, 0.41)

No overlap so predict favor school choice by as little as 2% or as much as 10%

Used n=2879 for MOE of () = 0.02

when should have used n=1000 for MOE of ( = 0.03

0.45+.03=> (0.42, 0.48)

0.39 +.03=> (0.36, 0.42)

CI’s overlap

Questions to ask

Who carried out the survey or poll?

What was the population?

How was the sample selected?

How large was the sample?

Sample size

Margin of error

What was the response rate?

How were the subjects contacted?

When was the survey conducted?

What were the exact questions?

What options were given for potential answers?

Issues

Population of Interest

Sampling Issues

Sample Selection

Undercoverage

Sample Size

Statistics ≠ Parameters

Non-sampling Issues

Processing Errors

Processing and Analysis Methods

Response Rate and Impact of Nonresponse

Methods Subjects Contacted

When Survey Conducted

Questions and Wording of Questions

Who Conducted Survey

Examples of Issues with Polls

Straw polls

http ://news.yahoo.com/iowa-straw-poll-outs-gop-establishment-080859714--election.html

Push poll examples in South Carolina

https :// www.washingtonpost.com/news/the-fix/wp/2016/02/12/the- south- carolina -push-poll-controversy-explained-video/

http ://www.sourcewatch.org/wiki.phtml?title=Push_poll

Sample Data

Qualitative

Quantitative

Parameter p

Point estimator

Point estimate value of value of

95% CI

range

10,000

0.01

MOE

range

MOE

range

MOE

range

MOE

range