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Chapter 14 Introduction to Inference Part B 1 Hypothesis Tests of Statistical “Significance” • Test a claim about a parameter • Often misunderstood • Has an elaborate vocabulary 2 4-step Process Hypothesis Testing Introduction to Inference 3 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? 4 Step 2: Plan 1. Identify the parameter (in this chapter we try to infer population mean µ) 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) Introduction to Inference 5 Null and alt hypotheses H0 & Ha • H0 = a claim of “no difference” or “no change” • Ha = a claim of “difference” or “change” • Ha can be one-sided or two-sided • One-sided Ha specifies the direction of the change (e.g., weight GAIN) • Two-sided Ha does not specify the direction of change (e.g., weight CHANGE, either increase or decrease) 6 Step 3: “Solve” has 3 sub-steps 1. Simple conditions (see prior lecture) a) SRS b) Normality c) σknown before collecting data 2. Calculate test statistics In this chapter “z Statistic” 3. Find P-value 7 Conditions • Data were collected via SRS • We know population standard deviation σ= 1 for weight change in current population • If H0 is true, then the sampling distribution of the mean based on n = 10 will be Normal with µ = 0 and 1 x 0.316 n 10 Introduction to Inference 8 What • In practice we rely on is the ? and to shed light on population shape may suggests a population is Normal • If individual data points are available, by making stemplot • The shape of the data (sample) should parallel the shape of the population. • Beware that small samples have a lot of chance variation. It is is difficult to judge “Normality” in small samples (but you can still check only for clear departures) 9 Test Statistic Standardize the sample mean zstat x 0 n Suppose: x-bar = 1.02, n = 10, σ= 1 x μ0 zstat σ n 1.02 0 1 10 3.23 X-bar is ~3 standard deviations greater than expected under H0 10 P-Value from Z Table If 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: μ μ0 P-value = 2×one-tailed P-value 11 P-value from Z Table • Draw • One-sided P-value = Pr(Z > 3.23) = 1 − .9994 = .0006 • Two-sided P-value = 2 × one-sided P = 2 × .0006 = .0012 12 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: Smaller-and-smaller Pvalues → stronger-and-stronger evidence against H0 • 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 13 “Significance Level” • α (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 rule P-value ≤ α evidence is significant P-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 14 Step 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). 15 Basics 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 17