Chapter 11: Inference About a Mean

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# Chapter 11: Inference About a Mean - PowerPoint PPT Presentation

Chapter 11: Inference About a Mean. In Chapter 11:. 11.1 Estimated Standard Error of the Mean 11.2 Student’s t Distribution 11.3 One-Sample t Test 11.4 Confidence Interval for μ 11.5 Paired Samples 11.6 Conditions for Inference 11.7 Sample Size and Power. σ not known.

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### In Chapter 11:

11.1 Estimated Standard Error of the Mean

11.2 Student’s t Distribution

11.3 One-Sample t Test

11.4 Confidence Interval for μ

11.5 Paired Samples

11.6 Conditions for Inference

11.7 Sample Size and Power

σ not known
• Prior chapter: σ was known before collecting data z procedures used to help infer µ
• When σ NOT known, calculate sample standard deviations sand use it to calculate this standard error:

The Normal distribution doesn’t fit well

• Using s instead of σ adds uncertainty to inferences  can NOT use z procedures
• Instead, rely on Student’s t procedures

William Sealy Gosset (1876–1937)

Student’s t distributions
• Familyof probability distributions
• Family members identified by degrees of freedom (df)
• Similar to “Z”, but with broader tails
• As df increases → tails get skinnier → t become like z

A t distribution with infinite degrees of freedom is a Standard Normal Z distribution

Table C (t table)

Rows  df

Columns  probabilities

Entries  t values

Notation: tcum_prob,df = t value

Example: t.975, 9 = 2.262

Objective: test a claim about population mean µ

Conditions :

Simple Random Sample

Normal population or “large sample”

One-Sample t Test
Hypothesis Statements
• Null hypothesisH0: µ = µ0

where µ0 represents the pop. mean expected by the null hypothesis

• Alternative hypotheses

Ha: µ < µ0(one-sided, left)

Ha: µ > µ0 (one-sided, right)

Ha: µ ≠ µ0 (two-sided)

Example
• Do SIDS babies have lower average birth weights than a general population mean µ of 3300 gms?
• H0: µ = 3300
• Ha: µ < 3300 (one-sided) or Ha: µ ≠ 3300 (two-sided)
One-Sample t Test Statistic

where

This t statistic has n – 1 degrees of freedom

Example (Data)

SRS n = 10 birth weights (grams) of SIDS cases

Example

Testing H0: µ = 3300

|tstat| =1.80

P-value via Table C
• Bracket |tstat| between t critical values
• For |tstat| = 1.80 with 9 df

Thus  One-tailed: 0.05 < P < 0.10

Two-tailed: 0.10 < P < 0.20

For a more precise P-value use a computer utility

Here’s output from the free utility StaTable

Graphically:

Interpretation
• TestingH0: µ = 3300 gms
• Two-tailed P > .10
• Conclude: weak evidence against H0
• The sample mean (2890.5) is NOT significantly different from 3300
Same Data

Interpretation: Population mean µ is between 2375 and 3406 grams with 95% confidence

§11.5 Paired Samples
• Two samples
• Each data point in one sample uniquely matched to a data point in the other sample
• Examples of paired samples
• “Pre-test/post-test”
• Cross-over trials
• Pair-matching
Example
• Does oat bran reduce LDL cholesterol?
• Start half of subjects on CORNFLK diet
• Start other half on OATBRAN
• Two weeks  LDL cholesterol
• Washout period
• Cross-over to other diet
• Two weeks  LDL cholesterol
Oat bran dataLDL cholesterol mmol

Subject CORNFLK OATBRAN

---- ------- -------

1 4.61 3.84

2 6.42 5.57

3 5.40 5.85

4 4.54 4.80

5 3.98 3.68

6 3.82 2.96

7 5.01 4.41

8 4.34 3.72

9 3.80 3.49

10 4.56 3.84

11 5.35 5.26

12 3.89 3.73

Within-pair difference “DELTA”
• Let DELTA = CORNFLK - OATBRAN
• First three observations in OATBRAN data:

ID CORNFLK OATBRAN DELTA ---- ------- ------- ----- 1 4.61 3.84 0.77

2 6.42 5.57 0.85

3 5.40 5.85 -0.45

etc.

All procedures are now directed toward difference variable DELTA

Exploratory and descriptive stats

DELTA: 0.77, 0.85, −0.45, −0.26, 0.30, 0.86, 0.60, 0.62, 0.31, 0.72, 0.09, 0.16

Stemplot

|-0f|4|-0*|2

|+0*|01|+0t|33

|+0f|

|+0s|6677

|+0.|88×1 LDL (mmol)

subscript d denotes “difference”

95% CI for µd

 95% confident population mean difference µd is between 0.105 and 0.656 mmol/L

Claim: oat bran diet is associated with a decline (one-sided) or change (two-sided) in LDL cholesterol.

Test H0: µd = µ0 where µ0 = 0 Ha: µd > µ0 (one-sided) Ha: µ ≠ µ0 (two-sided)

Hypothesis Test

|tstat| =3.043

P-value via Table C

Thus  One-tailed: .005 < P < .01

Two-tailed: .01 < P < .02

Interpretation

My P value is smaller than yours!

• Testing H0: µ = 0
• Two-tailed P = 0.011
•  Good reason to doubt H0
• (Optional) The difference is “significant” at α = .05 but not at α = .01
The Normality Condition
• t Procedures require Normal population or large samples
• How do we assess this condition?
• Guidelines. Use t procedures when:
• Population Normal
• population symmetrical and n ≥ 10
• population skewed and n≥ ~45(depends on severity of skew)
Can a t procedures be used?

Skewed small sample  avoid t procedures

Can a t procedures be used?

Mild skew in moderate sample t OK

Can a t procedures be used?

Skewed moderate sample  avoid t

Sample Size and Power

Methods:

(1) n required to achieve m when estimating µ

(2) n required to test H0 with 1−β power

(3) Power of a given test of H0

Power
• α≡ alpha (two-sided)
• Δ≡ “difference worth detecting” = µa – µ0
• n ≡ sample size
• σ≡ standard deviation
• Φ(z) ≡ cumulative probability of Standard Normal z score

with

.

Power: SIDS Example
• Let α = .05 and z1-.05/2 = 1.96
• Test: H0: μ = 3300 vs. Ha: μ = 3000. Thus:Δ≡ µ1 – µ0 = 3300 – 3000 = 300
• n = 10 and σ≡ 720 (see prior SIDS example)

Use Table B to look up cum prob  Φ(-0.64) = .2611