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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. Statistical inference.

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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
  • 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
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
Parameters and statistics

5: Intro to estimation

sampling distribution of the mean
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
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
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
95% Confidence Interval for µ

Formula for a 95% confidence interval for μ when σ is known:

5: Intro to estimation

illustrative example
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 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
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
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
(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
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
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
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 is not known
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
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
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
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 not known
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
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
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
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
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
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