Estimation Procedures: Constructing Confidence Intervals Using the Sampling Distribution, Slides of Statistics for Psychologists

Download Estimation Procedures: Constructing Confidence Intervals Using the Sampling Distribution and more Slides Statistics for Psychologists in PDF only on Docsity! Healey Chapter 7 Estimation Procedures Using the Sampling Distribution to Construct Confidence Intervals Docsity.com Outline: • The logic of estimation • How to construct and interpret confidence interval estimates for: – Sample means – Sample Proportions Docsity.com Logic (cont.) • Sampling Distribution is the link between sample and population. • The value of the parameters is unknown but characteristics of the Sampling Distribution are defined by theorems. POPULATION SAMPLING DISTRIBUTION SAMPLE Docsity.com Two Estimation Procedures • 1. A point estimate is a sample statistic used to estimate a population value: – The London Free Press reports that “42% of a sample of randomly selected city residents voted Liberal.” • 2. Confidence intervals (for means or proportions) consist of a range of values: – …”between 38% and 46% of city residents voted Liberal.” Docsity.com Bias and Efficiency • Bias: – An estimator of a mean (or a proportion) is unbiased if the mean of its sampling distribution is equal to the population mean. • Efficiency: – The smaller the standard error (S.D. of the sampling distribution,) the more the samples are clustered about the mean of the S.D. – This is known as efficiency. Docsity.com Confidence Levels (cont.) When α = .05… …then .025 of the area is distributed on either side (C ) The .95 in the middle section is our confidence level. The cut-off between our confidence level and +/- .025 is represented by a Z-value of +/- 1.96. c c Docsity.com Z-values for Various Alpha Levels Confidence Level α α/2 Z-score 90% .10 .0500 +/-1.65 95% .05 .0250 +/-1.96 99% .01 .0050 +/-2.58 99.9% .001 .0005 +/-3.29 (Note: Z-scores are found in Appendix A using the area for α/2) Docsity.com Confidence Intervals For Means Procedure: • 1. Set the alpha (the probability that the interval will be wrong). Note that the symbol for alpha is α. – Setting alpha equal to 0.05, a 95% confidence level, means the researcher is willing to be wrong 5% of the time. • 2. Find the Z-value associated with alpha. – If alpha is equal to 0.05, we would place half (0.025) of this probability in the lower tail and half in the upper tail of the distribution. • 3. Substitute values into formula and solve. Formula: c.i. =         − Ζ±Χ 1N s Docsity.com Example (cont.) • We can estimate that households in this community average 6.0 ± .44 hours of TV watching each day. • Another way to state the interval: 5.56 ≤ μ ≤ 6.44 Interpretation: We estimate, with 95% confidence, that the population mean for TV watching is greater than or equal to 5.56 and less than or equal to 6.44. (This interval has a .05 chance of being wrong.) Docsity.com Example (cont.) • In other words: • Even if the statistic is as much as ±1.96 standard deviations from the mean of the sampling distribution the confidence interval will still include the value of μ. • Only rarely (5 times out of 100) will the interval not include μ. Docsity.com Confidence Intervals For Proportions • Procedure: – Set alpha = .05. – Find the associated Z score. – Substitute the sample information into formula: c.i. = Note: Ρs = sample proportion Ρu (when population proportion is not known,) is set to .50 ( ) Ν Ρ−Ρ Ζ±Ρ uus 1 Docsity.com Confidence Intervals For Proportions • Changing back to %, we estimate that 42% ± 4% of the city residents vote Liberal. • Another way to state the interval: 38% ≤ Pu ≤ 46% Interpretation: We estimate that the population value is greater than or equal to 38% and less than or equal to 46% for city residents who vote Liberal. (This interval has a .05 chance of being wrong.) Docsity.com Practice Questions: • Healey 8e and 1st Cdn #7.5, 7.7, 7.9 • Healey 2nd Cdn #6.5, 6.7, 6.9 Docsity.com