5: Introduction to estimation

5: Introduction to estimation Intro to statistical inference Sampling distribution of the mean Confidence intervals (σ known) Student’s t distributions Confidence intervals (σ not known) Sample size requirements 5: Intro to estimation

Statistical inference • Statistical inference  generalizing from a sample to a population with calculated degree of certainty • Two forms of statistical inference • Estimation  introduced this chapter • Hypothesis testing  next chapter 5: Intro to estimation

Parameters and estimates • Parameter numerical characteristic of a population • Statistics = a value calculated in a sample • Estimate  a statistic that “guesstimates” a parameter • Example: sample mean “x-bar” is the estimator of population mean µ Parameters and estimates are related but are not the same 5: Intro to estimation

Parameters and statistics 5: Intro to estimation

Sampling distribution of the mean • x-bar takes on different values with repeated (different) samples • µ remain constant • Even though x-bar is variable, it’s “behavior” is predictable • The behavior of x-bar is predicted by its sampling distribution, the Sampling Distribution of the Mean (SDM) 5: Intro to estimation

Simulation experiment • Distribution of AGE in population.sav (Fig. right) • N = 600 • µ = 29.5 (center) • s = 13.6 (spread) • Not Normal (shape) • Conduct three sampling simulations • For each experiment • Take multiple samples of size n • Calculate means • Plot means  simulated SDMs • Experiment A: each sample n = 1 • Experiment B: each sample n = 10 • Experiment C: each sample n = 30 5: Intro to estimation

Results of simulation experiment • Findings: • SDMs are centered on 29 (µ) • SDMs become tighter as n increases • SDMs become Normal as the n increases 5: Intro to estimation

95% Confidence Interval for µ Formula for a 95% confidence interval for μ when σ is known: 5: Intro to estimation

Illustrative example • Example • Population with σ = 13.586 (known ahead of time) • SRS  {21, 42, 11, 30, 50, 28, 27, 24, 52} • n = 10, x-bar = 29.0 • SEM = s / n = 13.586 / 10 = 4.30 • 95% CI for µ = = xbar ± (1.96)(SEM) = 29.0 ± (1.96)(4.30) = 29.0 ± 8.4 = (20.6, 37.4) Margin of error 5: Intro to estimation

Margin of error • Margin or error d = half the confidence interval • Surrounded x-bar with margin of error • 95% CI for µ = xbar ± (1.96)(SEM) = 29.0 ± (1.96)(4.30) = 29.0 ± 8.4 point estimate margin of error 5: Intro to estimation

Interpretation of a 95% CI We are 95% confident the parameter will be captured by the interval. 5: Intro to estimation

Other levels of confidence Let a the probability confidence interval will not capture parameter 1 – athe confidence level 5: Intro to estimation

(1 – a)100% confidence for μ Formula for a (1-α)100% confidence interval for μ when σ is known: 5: Intro to estimation

Example: 99% CI, same data • Same data as before • 99% confidence interval for µ = x-bar ± (z1–.01/2)(SEM) = x-bar ± (z.995)(SEM) = 29.0 ± (2.58)(4.30) = 29.0 ± 11.1 = (17.9, 40.1) 5: Intro to estimation

Confidence level and CI length p. 5.9 demonstrates the effect of raising your confidence level  CI length increases  more likely to capture µ * CI length = UCL – LCL 5: Intro to estimation

Beware • Prior CI formula applies only to • SRS • Normal SDMs • σ known ahead of time • It does not account for: • GIGO • Poor quality samples (e.g., due to non-response) 5: Intro to estimation

When σ is Not Known • In practice we rarely know σ • Instead, we calculate s and use this as an estimate of σ • This adds another element of uncertainty to the inference • A modification of z procedures called Student’s t distribution is needed to account for this additional uncertainty 5: Intro to estimation

Student’s t distributions Brilliant! • William Sealy Gosset (1876-1937) worked for the Guinness brewing company and was not allowed to publish • In 1908, writing under the the pseudonym “Student” he described a distribution that accounted for the extra variability introduced by using s as an estimate of σ 5: Intro to estimation

t Distributions • Student’s t distributions are like a Standard Normal distribution but have broader tails • There is more than one t distribution (a family) • Each t has a different degrees of freedom (df) • As df increases, t becomes increasingly like z 5: Intro to estimation

t table • Each row is for a particular df • Columns contain cumulative probabilities or tail regions • Table contains t percentiles (like z scores) • Notation: tdf,p Example: t9,.975 = 2.26 5: Intro to estimation

95% CI for µ, σ not known Formula for a (1-α)100% confidence interval for μ when σ is NOT known: Same as z formula except replace z1-a/2 with t1-a/2 and SEM with sem 5: Intro to estimation

Illustrative example: diabetic weight • To what extent are diabetics over weight? • Measure “% of ideal body weight” = (actual body weight) ÷ (ideal body weight) × 100% • Data (n = 18):{107, 119, 99, 114, 120, 104, 88, 114, 124, 116, 101, 121, 152, 100, 125, 114, 95, 117} 5: Intro to estimation

Interpretation of 95% CI for µ • Remember that the CI seeks to capture µ, NOT x-bar • 95% confidence means that 95% of similar intervals would capture µ (and 5% would not) • For the diabetic body weight illustration, we can be 95% confident that the population mean is between 105.6 and 120.0 5: Intro to estimation

Sample size requirements • Assume: SRS, Normality, valid data • Let d  the margin of error (half confidence interval length) • To get a CI with margin of error ±d, use: 5: Intro to estimation

Sample size requirements, illustration Suppose, we have a variable with s= 15 Smaller margins of error require larger sample sizes 5: Intro to estimation

Acronyms SRS  simple random sample SDM  sampling distribution of the mean SEM  sampling error of mean CI  confidence interval LCL  lower confidence limit UCL  lower confidence limit 5: Intro to estimation

5: Introduction to estimation