Chapter 14 Introduction to Inference 1

Chapter 14

Introduction to Inference

What is Statistical InferenceStatistical Inference?For everyone who does habitually attempt the difficult task of making sense of figures is, in fact, assaying a logical process of the kind we call induction, in that he is attempting to draw draw

inferencesinferences from the from the particular to the generalparticular to the general; or, as we more usually say in statistics, from the sample to the population.

R.A. Fisher (1890 – 1962)Father of modern statistics

Chapter 14 Introduction to Inference 3

Two forms of statistical inference:Two forms of statistical inference:• EstimationEstimation (Confidence Intervals)(Confidence Intervals)

• Hypothesis Tests Hypothesis Tests (Significance)(Significance)

Statistical Inference

Chapter 14 Introduction to Inference 4

Statistical Inference• Objective to infer parametersparameters• ParameterParameter ≡ a numerical characteristic of a

population or probability function• Examples of parameters:parameters:

μ (population mean; expected value)σ (population standard deviation; standard dev

parameter)p (probability of “success,” population proportion)

• Chs 14 & 15 introduces conceptsconcepts about inference• Chs 15–20 introduces inferential techniques techniques

Chapter 14 Introduction to Inference 5

“Simple Conditions” for Chapter 14

• Data acquired by simple random sample (SRS), i.e., all potential observations have same probability of entering the sample

• No major deviations departures from Normality in population

• Value of σ is known or assumed before collecting data

Objective: to infer μ!

Chapter 14 Introduction to Inference 6

Example Example ““Female BMIFemale BMI””• Statement: What is the mean BMI

µ in females between ages 20 and 29?

• Body Mass Index Body Mass Index ≡ BMI = BMI = weight / heightweight / height22

• Assume “simple conditions”1. SRS 2. Population approx. normal3. σ = 7.5 (assumed before data collected)

• Plan: Estimate µ with 95% confidence

Chapter 14 Introduction to Inference 7

Reasoning behind estimationReasoning behind estimation• If I took a multiple SRSs, the sample means (x-bars)

would be different in each one.• We do not expect x-bar to be exactly equal to µ any

given x-bar is just an estimate of µ.• The variability of the x-bars in predictable in the form of

a sampling distribution of meanssampling distribution of means• Fact: Fact: Under the “simple conditions” in this chapter, the

sampling distributions of meanssampling distributions of means will be Normal distribution with mean µ and standard deviation:

n

x

← Standard Deviation of the Mean(also referred to as the standard errorof the mean)

Chapter 14 Introduction to Inference 8

In our example, n = 654 and σ = 7.5. Therefore: Example (Female BMI)Example (Female BMI)

0.3 (rounded)7.5

654x

n

• σx-bar tells us how closehow close x-bar is likely to be to µ

• The 68-95-99.7 rule tells us that x-bar will be within two two σσx-barx-bar units (that’s 0.6) of µ in 95% of samples

If we say that µ lies in the interval (x-bar − 0.6) to (x-bar + 0.6), wewe’’ll be right 95% of the time ll be right 95% of the time

• Therefore, we can be 95% confident 95% confident that an interval “x-bar ± 0.6” will capture µ

Chapter 14 Introduction to Inference 9

Basis of CBasis of Confidence IIntervals (CIsCIs)

Chapter 14 Introduction to Inference 10

CConfidence IInterval (CICI)

• The CICI has two parts

point estimate point estimate ± margin of error± margin of error• Suppose in our particular sample, the mean is

26.8. This is the point estimatepoint estimate for µ.• Recall from the previous slide that the margin of margin of

errorerror for our data is 0.6 (with 95% confidence)• Therefore, the 95% confidence interval95% confidence interval (for this

particular sample) = 26.8 ± 0.6 = (26.2, 27.4).

Chapter 14 Introduction to Inference 11

CConfidence Level Level CC

• CIsCIs can be calculated at different levels of levels of confidence.confidence.

• Let Let CC represent the probability the interval will capture the parameter

• In our example, C = 95%• Other common levels of confidenceOther common levels of confidence are 90%

and 99%.• In this chapter we adjust the C level by changing

the z* critical valuez* critical value.

Chapter 14 Introduction to Inference 12

Confidence Levels & Confidence Levels & zz critical values critical valuesIn this Chapter we adjust the confidence level by altering critical value z*critical value z*

Common levels of confidence & z critical values

Confidence level C 90% 95% 99%

Critical value z* (table Ctable C) 1.645 1.960 2.576

Chapter 14 Introduction to Inference 13

C level CI for μ, σ known“z procedure”

To estimate µ with confidence level C, use

x zn

Use Table C to determine value of z*

Chapter 14 Introduction to Inference 14

Example Example (95% CI): Solve & Conclude(95% CI): Solve & Conclude

x zn

26.8 0.6

(26.2, 27.4)

Conclude: We are 95% confident population mean BMI µ is between 26.0 and 27.6

7.526.8 (1.960)

654

Data: 654, 26.8n x

Chapter 14 Introduction to Inference 15

Now have students calculate a 99% Now have students calculate a 99% CI with the dataCI with the data

x zn

Conclude: We are 99% confident population mean BMI µ is between “lower confidence limit (LCL) here” and “upper confidence limit (UCL) here.”

Data: 654, 26.8n x

Hint: The only thing that changes is the z* critical value.

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Interpreting a CI

• Confidence levelConfidence level CC is the success rate of the method that produced the interval.

• We knowWe know with C level of confidence that the CI will capture µ.

• We donWe don’’t knowt know with certainty whether any given CI will capture µ or missed it.

Chapter 14 Introduction to Inference 17

Four-Step Procedure for CIs

Stopping Point for Exam 2

Slides after this point forward could be edited after exam 2

Chapter 14 Introduction to Inference 19

Hypothesis Hypothesis (“Significance”) (“Significance”)

TestsTests• Objective Objective test a claim about a

parameter

• Uses an elaborate vocabularyelaborate vocabulary

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4-step Process 4-step Process Hypothesis (Significance) TestingHypothesis (Significance) Testing

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State and PlanState and PlanExample Example ““Population Weight Gain?Population Weight Gain?””StateState: Is there good evidence that the

population is gaining weight?

PlanPlan• ParameterParameter is population mean

weight gain µ• Null hypothesis HNull hypothesis H00 statement

of “no difference” population not gaining weight H0: μ = 0

• Alternative hypothesis Alternative hypothesis HHaa population gaining weight Ha: μ > 0

• Type of testType of test: z test if “simple conditions” (slide 5) met

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Notes on Statistical Hypotheses• H0 is key to understanding

• Ha contradicts H0

• Ha can be stated in one-sided or two-sided ways

– One-sided One-sided HHaa specifies the direction of the difference weight GAIN in population Ha: μ > 0

– Two-sided Two-sided HHaa does not specific the direction of the difference weight CHANGE in the population Ha: μ ≠ 0

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Example Example ““Weight GainWeight Gain” ”

“Solve” Sub-steps “Solve” Sub-steps (a) Check conditionsconditions

SRSSRS

No major departures from NormalityNormality

σknownσknown before collecting data

(b) Calculate Calculate statistics See “z Statistic” Slide

(c) Find PP-value-value

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ReasoningReasoning of Significance Testing• IfIf H0 and the conditions are

true, then the sampling distribution of x-bar would be Normal with µ = 0 and

• IfIf a study produced an x-bar of 0.3, this would be poor evidence against H0

• IfIf a different study produced an x-bar of 1.02, this would be good evidence against H0

316.010

1

nx

~ (0,0.316)x N

Chapter 14 Introduction to Inference 2525

Test Statistic Standardize the sample mean

stat

x μ

zσ

n

0

0 statx

zn

X-bar is 3 standard deviations greater than expected if H0 true

Suppose: x-bar = 1.02, n = 10, and σ = 1

1.02 0

110

3.23

Chapter 14 Introduction to Inference 2626

P-Value from Z TableZ TableFor Ha: μ> μ0

P-value = Pr(Z > zstat)

= right-tail beyond zstat

• For Ha: μ< μ0

P-value = Pr(Z < zstat)

= left tail beyond zstat

• For Ha: μμ0

P-value = 2 × one-tailed P-value

Chapter 14 Introduction to Inference 2727

P-value from Z Table Z Table • Draw (right)Draw (right) • One-sided P-value

= Pr(Z > 3.23) = 1 − .9994 = .0006

• Two-sided P-value = 2 × one-sided P = 2 × .0006 = .0012

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P-value: Interpretation • P-value ≡ the probability the data would take a value as

extreme or more extreme than observed if H0 were true

• Smaller-and-smaller Smaller-and-smaller PP-values → stronger-and--values → stronger-and-stronger evidence stronger evidence against Hagainst H00

• Conventions

.10 < P < 1.0 insignificant evidence against H0

.05 < P ≤ .10 marginally significant evidence vs. H0

.01 < P ≤ .05 significant evidence against H0

0 < P ≤ .01 highly significant evidence against H0

Chapter 14 Introduction to Inference 3030

• αα (alpha) ≡ threshold for “significance”• If we choose α = 0.05, we require evidence so

strong that it would occur no more than 5% of the time when H0 is true

• Decision ruleDecision ruleP-value ≤ α evidence is significantP-value > α evidence not significant

• For example, let α = 0.01. The two-sided P-value = 0.0012 is less than .01, so data are significant at the α = .01 level.

““Significance LevelSignificance Level””

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Example Example ““Weight GainWeight Gain” ”

ConclusionConclusion• The P-value of .0012 provides highly significant

evidence against H0: µ = 0

We rule in favor of Ha: µ ≠ 0

• Conclude: the population’s mean weight in changing

• Our sample mean weight gain of 1.02 pounds per person is statistically significant at the α= .002 level but not at the α= .001 level

Chapter 14 Introduction to Inference 32Basics of Significance Testing 32