5: Introduction to estimation

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# 5: Introduction to estimation - PowerPoint PPT Presentation

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

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)
• 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