October 15

In Chapter 9: 9.1 Null and Alternative Hypotheses 9.2 Test Statistic 9.3 P-Value 9.4 Significance Level 9.5 One-Sample z Test 9.6 Power and Sample Size

Review: Basics of Inference Population  all possible values Sample  a portion of the population Statistical inference  generalizing from a sample to a population with calculated degree of certainty Two forms of statistical inference –Hypothesis testing –Estimation Parameter  a numerical characteristic of a population, e.g., population mean µ, population proportion p Statistic  a calculated value from data in the sample, e.g., sample mean ( ), sample proportion ( )

Distinctions Between Parameters and Statistics (Chapter 8 review) ParametersStatistics SourcePopulationSample Notation Greek (e.g.,  ) Roman (e.g., xbar) VaryNoYes CalculatedNoYes

Ch 8 Review: How A Sample Mean Varies Sampling distributions of means tend to be Normal with an expected value equal to population mean µ and standard deviation (SE) = σ/√n

Introduction to Hypothesis Testing Hypothesis testing is also called significance testing The objective of hypothesis testing is to test claims about parameters The first step is to state the problem! Then, follow these four steps --

Hypothesis Testing Steps A.Null and alternative hypotheses B.Test statistic C.P-value and interpretation D.Significance level (optional) Understand all four steps of testing, not just the calculations.

§9.1 Null and Alternative Hypotheses Convert the research question to null and alternative hypotheses The null hypothesis (H 0 ) is a claim of “no difference in the population” The alternative hypothesis (H a ) is a claim of “difference” Seek evidence against H 0 as a way of bolstering H a

Illustrative Example: “Body Weight” Statement of the problem: In the 1970s, 20–29 year old men in the U.S. had a mean body weight μ of 170 pounds. Standard deviation σ = 40 pounds. We want to test whether mean body weight in the population now differs. H 0: μ = 170 (“no difference”) The alternative hypothesis can be stated in one of two ways: H a: μ > 170 (one-sided alternative) H a: μ ≠ 170 (two-sided alternative)

§9.2 Test Statistic This chapter introduces the z statistic for one- sample problems about means.

Illustrative Example: z statistic For the illustrative example, μ 0.= 170 We take an SRS of n = 64 and know We know that σ = 40 so the standard error of the mean is If we found a sample mean of 173, then If we found a sample mean of 185, then

Reasoning Behind the z stat This is the sampling distribution of the mean when μ = 170, σ = 40, and n = 64.

§9.3 P-value The P-value answer the question: What is the probability of the observed a test statistic equal to or one more extreme than the current statistic assuming H 0 is true? This corresponds to the AUC in the tail of the Z sampling distribution beyond the z stat. Use Table B or a software utility to find this AUC. (See next slide). Smaller and smaller P-values provider stronger and stronger evidence against H 0

The one-sided P-value for z stat = 0.6 is

The one-sided P-value for z stat = 3.0 is

Two-Sided P-Value For one-sided H a  use the area in the tail beyond the z statistic For two-sided H a  consider deviations “up” and “down” from expected  double the one-sided P-value For example, if the one-sided P-value = , then the two-sided P-value = 2 × = If the one-sided P = , then the two-sided P = 2 × =

Two-tailed P-value

§9.4 Significance Level Smaller and smaller P-values provide stronger and stronger evidence against H 0 :Although it unwise to draw firm cutoffs, here are conventions that used as a starting point: P > 0.10  non-significant evidence against H < P  0.10  marginally significant against H < P  0.05  significant evidence against H 0 P  0.01  highly significant evidence against H 0 Examples P =.27 is non-significant evidence against H 0 P =.01 is highly significant evidence against H 0

Decision Based on α Level Let α represent the probability of erroneously rejecting H 0 Set α threshold of acceptable error (e.g., let α =.10, let α =.05, or whatever level is acceptable) Reject H 0 when P ≤ α Example: Set α =.10. Find P = Since P > α, retain H 0 Example: Set α =.05. Find P =.01. Since P < α reject H 0

§9.5 One-Sample z Test (Summary) Test conditions: Quantitative response Good data SRS (or facsimile) σ known (not calculated) Population approximately Normal or sample large (central limit theorem) Test procedure A.H 0 : µ = µ 0 vs. H a : µ ≠ µ 0 (two-sided) or H a : µ µ 0 (right-sided) B.Test statistic C.P-value: convert z stat to P value [Table B or software] D.Significance level (optional)

Example: The “Lake Wobegon Problem” Typically, Weschler Adult Intelligence Scores are Normal with µ = 100 and  = 15 Take an SRS of n = 9 Measure scores  {116, 128, 125, 119, 89, 99, 105, 116, 118} Calculate x-bar = Does this sample mean provide statistically reliable evidence that population mean μ is greater than 100?

Illustrative Example: Lake Wobegon A.Hypotheses: H 0 : µ = 100 H a : µ > 100 (one-sided) B.Test statistic:

C. P-value: Pr(Z ≥ 2.56) = (Table B) D. Significance level (optional): This level of evidence is significant at α 0.01 (reject H 0 ). P =.0052  highly significant evidence against H 0

Two-Sided Alternative (Lake Wobegon Illustration) Two-sided alternative H a : µ ≠100 doubles P = 2 × = P =.0104 provides significant against H 0

§9.6 Power and Sample Size Truth Decision H 0 trueH 0 false Retain H 0 Correct retentionType II error Reject H 0 Type I errorCorrect rejection α ≡ probability of a Type I error β ≡ Probability of a Type II error Two types of decision errors: Type I error = erroneous rejection of a true H 0 Type II error = erroneous retention of a false H 0

Power The traditional hypothesis testing paradigm considers only Type I errors However, we should also consider Type II errors β ≡ probability of a Type II error 1 –  “Power” ≡ probability of avoiding a Type II error

Power of a z test where Φ(z) represent the cumulative probability of Standard Normal z (e.g., Φ(0) = 0.5) μ 0 represent the population mean under the null hypothesis μ a represents the population mean under an alternative hypothesis with.

Calculating Power: Example A study of n = 16 retains H 0 : μ = 170 at α = 0.05 (two-sided). What was the power of the test conditions to identify a significant difference if the population mean was actually 190?

Reasoning Behind Power Consider two competing sampling distribution models –One model assumes H 0 is true (top curve, next page) –An other model assumes H a is true μ a = 190 (bottom curve, next page) When α = 0.05 (two-sided), we will reject H 0 when the sample mean exceeds (right tail, top curve) The probability of getting a value greater than on the bottom curve is , corresponding to the power of the test

Sample Size Requirements The required sample size for a two-sided z test to achieve a given power is where 1 – β ≡ desired power of the test α ≡ desired significance level σ ≡ population standard deviation Δ = μ 0 – μ a ≡ the difference worth detecting

Illustrative Example: Sample Size Requirement How large a sample is needed for a one-sample z test with 90% power and α = 0.05 (two-tailed) when σ = 40. The null hypothesis assumes μ = 170 and the alternative assumes μ = 190. We look for difference Δ = μ 0 − μ a = 170 – 190 = −20 Round this up to 42 to ensure adequate power.

Example showing the conditions for 90% power.