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Transcript of Chapter 14Introduction to Inference1 Chapter 14 Introduction to Inference.
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.
Chapter 14 Introduction to Inference 16
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
Chapter 14 Introduction to Inference 20
4-step Process 4-step Process Hypothesis (Significance) TestingHypothesis (Significance) Testing
Chapter 14 Introduction to Inference 21
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
Chapter 14 Introduction to Inference 2222
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
Chapter 14 Introduction to Inference 2323
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
Chapter 14 Introduction to Inference 2424
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
Chapter 14 Introduction to Inference 2929
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””
Chapter 14 Introduction to Inference 3131
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