Confidence Intervals & Polls Lesson 7

Estimating m Sample mean is a point estimate of m Single value SE as a measure of fit Confidence intervals Boundaries for true value of m 95% most common; 99% also used Based on z or t Mean & standard error ~

Computing Confidence Intervals When s is known Mainly for standardized variables e.g., IQ, ACT, SAT, GRE General formula Use z scores that at boundaries of desired interval

95% Confidence Intervals for IQ Sample from population of IQ scores z = ~

95% Confidence Intervals for IQ Lower boundary Upper boundary or How does interval change if n=25? Wider or narrower? Why? What does it mean? ~

Changing the Confidence Level 99% confidence interval z = How does interval change? ~ Lower boundary Upper boundary

Meaning of Confidence Interval E.g., 95% confidence interval If we compute confidence intervals for 100 samples 95 would contain true value of m 5 would not No way to know which group any single confidence interval falls in Always some probability it does not P (error) = .05 ~

Confidence Intervals: When s unknown Same general procedure & formula Differences Must use s instead of s Introduces more uncertainty (error) Use estimated standard error Use t instead of z df important ~

The t distribution 95% confidence intervals More uncertainty  need wider intervals t distribution Adjusts based on sample size df = n-1 Very large samples: t ≈ z Smaller samples: t > z Critical values of the t-distribution A.2, pg 803 ~

The t distribution To find t, need… df = n-1 Confidence level Actually area in tails (1-p) Use Two-tailed Test column Law of large numbers As n  Uncertainty  ~

Computing confidence Intervals with t s is unknown Need t value df = n-1; 1-p Estimated standard error ~

99% Confidence Interval: s unknown Mean number hours spent studying for a psychology course? ~

99% Confidence Interval: s unknown Lower boundary Upper boundary or How does interval change if… Smaller or larger sample? 95% confidence level? ~

Opinion Polls & Confidence Intervals Opinion polls ubiquitous Politics, public policy, consumer products, etc. Often misrepresented / misinterpreted Poll results reported as Proportion & margin of error Actually reporting confidence interval ~

Opinion Polls in the News Give example of poll Changes from week to week Can we say things are really changing?

Opinion Polls & Confidence Intervals Our example: Binomial data only 2 possible responses Yes/no; Candidate A or Candidate B Proportions Proportion choosing a response Parameter: P, statistic: p All responses: p + (1-p) Margin of error ~

Computing Margin of Error Confidence interval formula Computing standard error of proportion Rest of computation same as before ~

Example: Election Opinion Poll 250 voters asked if they will vote for… Jane Jones (55%) John Smith (45%) At 95% confidence level What is margin of error? Can we say Ms. Jones will win? ~

Example: Computing Margin of Error Compute standard error Compute margin of error

Example: 95% Confidence Intervals Jane Jones John Smith If Smith wins were polls wrong? No, within margin of error. ~

A Guide for Interpreting Polls Who paid for poll and conducted it? Do they have vested interest? Are they reputable / capable? How many participants & how chosen? Random sample? Important subgroups? Ability to give informed answers? e.g., 1st graders on national elections? Opinions represent only group actually polled e.g., college-age views may not be representative of all adults ~ Adapted from article by Ken Blake, PhD, MTSU School of Journalism

A Guide for Interpreting Polls The wording & order of questions Both can bias responses The confidence level & margin of error Margin of error must be recalculated if drawing conclusions about subgroups What is the response rate? Low rates  respondents may be very different from non-respondents ~ Adapted from article by Ken Blake, PhD, MTSU School of Journalism

Confidence Intervals Can be computed for many statistics Formula for standard error changes Related to hypothesis testing Also alternative to it, along with meta-analysis Covered in next lesson ~