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Chapter 14Introduction to Inference

Part BPart B

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HypothesisHypothesis TestsTests of of Statistical Statistical ““SignificancSignificanc

ee””• Test a claim about a parameter • Often misunderstood• Has an elaborate vocabularyelaborate vocabulary

Introduction to Inference 3

4-step Process 4-step Process Hypothesis TestingHypothesis Testing

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Step 1: StatementStep 1: Statement• The illustrative example will address

whether people in a population are gaining weight

• We collect an SRS of n = 10 individuals from the population and determine the mean weight change in the sample (x-bar)

• At what point do we declare that an observed increase “statistically significant” and applies to the entire population?

Introduction to Inference 5

Step 2: PlanStep 2: Plan1.1. Identify the parameterIdentify the parameter (in

this chapter we try to infer population mean µ)

2. State the null and alternative null and alternative hypotheseshypotheses (next slide)

3. Determine what test is appropriate (in this chapter we use the one-sample z test)

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Null and alt hypotheses HH0 0 && HHaa

• H0 = a claimclaim of “no difference” or “no change”

• Ha = a claimclaim of “difference” or “change”

• Ha can be one-sided one-sided or two-sided two-sided • One-sided One-sided HHaa specifies the direction of

the change (e.g., weight GAIN)• Two-sided Two-sided HHaa does not specify the

direction of change (e.g., weight CHANGE, either increase or decrease)

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Step 3:Step 3: ““SolveSolve”” has 3 sub-steps has 3 sub-steps

1. Simple conditions (see prior lecture)conditions (see prior lecture)a) SRS a) SRS b) Normalityb) Normalityc) c) σknownσknown before collecting data

2.2. CalculateCalculate test statistics In this chapter “z Statistic”

3. Find PP-value-value

Introduction to Inference 88

Conditions• Data were collected via SRS Data were collected via SRS • We know population standard We know population standard

deviation deviation σσ= 1 for weight = 1 for weight change in current population change in current population

• IfIf H0 is true, then the samplingsampling

distribution of the meandistribution of the mean based on n = 10 will be Normal with µ = 0 and

316.0101

nx

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9

1010

Test Statistic Standardize the sample mean

stat

x μ

zσ

n

0

0 statx

zn

X-bar is ~3 standard deviations greater than expected under H0

Suppose: x-bar = 1.02, n = 10, σ= 1

1.02 01

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3.23

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

P-value = Pr(Z > zstat) = right-tail beyond zstat

If Ha: μ< μ0 P-value = Pr(Z < zstat) = left tail beyond zstat

If Ha: μμ0 P-value = 2×one-tailed P-value

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P-value from Z Table Z Table • DrawDraw• 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 ≡ probability data would take a value as extreme

or more extreme than observed data when H0 is true• Measure of evidence: Measure of evidence: Smaller-and-smaller Smaller-and-smaller PP--

values → stronger-and-stronger evidence values → stronger-and-stronger evidence against Hagainst H00

• ConventionsConventions.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

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

strong that a false rejection 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, the two-sided P-value of 0.0012 is significant at α = .002 but not at α = .001

““Significance LevelSignificance Level””

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Step 4: ConclusionStep 4: Conclusion• The P-value of .0012 provides “highly significant”

evidence against H0: µ = 0• Conclude: The sample demonstrates a

significant increase in weight (mean weight gain = 1.02 pounds, P = .0012).

16Basics of Significance Testing 16

Take out pencil and calculator

• Reconsider the weight change example• Recall σ= 1• Now take a different sample of n = 10 • This new sample has a mean of 0.3 lbs• Carry out the four-step solution to test

whether the population is gaining weight based on this new data

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