1 Chapter 14 Introduction to Inference Part B

2 Hypothesis Tests of Statistical “Significance” Test a claim about a parameter Often misunderstood elaborate vocabularyHas an elaborate vocabulary

Introduction to Inference3 4-step Process Hypothesis Testing 3

4 Step 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 Inference5 Step 2: Plan 1.Identify the parameter 1.Identify the parameter (in this chapter we try to infer population mean µ) null and alternative hypotheses 2.State the null and alternative hypotheses (next slide) 3.Determine what test is appropriate (in this chapter we use the one-sample z test) 5

66 H 0 & H a Null and alt hypotheses H 0 & H a claimH 0 = a claim of “no difference” or “no change” claimH a = a claim of “difference” or “change” one-sided two-sidedH a can be one-sided or two-sided One-sided H aOne-sided H a specifies the direction of the change (e.g., weight GAIN) Two-sided H aTwo-sided H a does not specify the direction of change (e.g., weight CHANGE, either increase or decrease)

7 7 Step 3: “Solve” has 3 sub-steps conditions (see prior lecture) 1.Simple conditions (see prior lecture) a) SRS b) Normality c) σknown c) σknown before collecting data 2.Calculate 2.Calculate test statistics In this chapter  “z Statistic” P-value 3.Find P-value

Introduction to Inference8 8 Conditions Data were collected via SRSData were collected via SRS We know population standard deviation σ= 1 for weight change in current populationWe know population standard deviation σ= 1 for weight change in current population Ifsampling distribution of the meanIf H 0 is true, then the sampling distribution of the mean based on n = 10 will be Normal with µ = 0 and 8

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10 Test Statistic Standardize the sample mean  X-bar is ~3 standard deviations greater than expected under H 0 Suppose: x-bar = 1.02, n = 10, σ= 1

11 Z Table P-Value from Z Table If H a : μ  > μ 0 P-value = Pr(Z > z stat ) = right-tail beyond z stat If H a : μ  < μ 0 P-value = Pr(Z < z stat ) = left tail beyond z stat If H a : μ  μ 0 P-value = 2×one-tailed P-value

12 Z Table P-value from 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

13 P-value: Interpretation P-value ≡ probability data would take a value as extreme or more extreme than observed data when H 0 is true Measure of evidence: Smaller-and-smaller P- values → stronger-and-stronger evidence against H 0Measure of evidence: Smaller-and-smaller P- values → stronger-and-stronger evidence against H 0 ConventionsConventions.10 < P < 1.0  insignificant evidence against H 0.05 < P ≤.10  marginally significant evidence vs. H 0.01 < P ≤.05  significant evidence against H 0 0 < P ≤.01  highly significant evidence against H 0

14 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 H 0 is true Decision ruleDecision rule P-value ≤ α  evidence is significant P-value > α  evidence not significant For example, the two-sided P-value of is significant at α =.002 but not at α =.001 “Significance Level”

15 Step 4: Conclusion The P-value of.0012 provides “highly significant” evidence against H 0 : µ = 0 Conclude: The sample demonstrates a significant increase in weight (mean weight gain = 1.02 pounds, P =.0012).

16Basics of Significance Testing16

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 17