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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

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

not known
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:
additional uncertainty
Additional Uncertainty

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
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
Table C (t table)

Rows  df

Columns  probabilities

Entries  t values

Notation: tcum_prob,df = t value

Example: t.975, 9 = 2.262

one sample t test
Objective: test a claim about population mean µ

Conditions :

Simple Random Sample

Normal population or “large sample”

One-Sample t Test
hypothesis statements
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
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
One-Sample t Test Statistic

where

This t statistic has n – 1 degrees of freedom

example data
Example (Data)

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

example12
Example

Testing H0: µ = 3300

p value via table c

|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
For a more precise P-value use a computer utility

Here’s output from the free utility StaTable

Graphically:

interpretation
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
Same Data

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

11 5 paired samples
§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
example19
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 data ldl cholesterol mmol
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
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
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 d
95% CI for µd

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

hypothesis test
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
p value via table c26

|tstat| =3.043

P-value via Table C

Thus  One-tailed: .005 < P < .01

Two-tailed: .01 < P < .02

interpretation29
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
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
Can a t procedures be used?

Skewed small sample  avoid t procedures

can a t procedures be used32
Can a t procedures be used?

Mild skew in moderate sample t OK

can a t procedures be used33
Can a t procedures be used?

Skewed moderate sample  avoid t

sample size and power
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
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
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