September 15

In Chapter 11: 11.1 Estimated Standard Error of the Mean 11.2 Student’s t Distribution 11.3 One-Sample t Test 11.4 Confidence Interval for μ 11.5 Paired Samples 11.6 Conditions for Inference 11.7 Sample Size and Power

§11.1 Estimated Standard Error of the Mean We rarely know population standard deviation σ  instead, we calculate sample standard deviations s and use this as an estimate of σ We then use s to calculate this estimated standard error of the mean: Using s instead of σ adds a source of uncertainty  z procedures no longer apply  use t procedures instead

§11.2 Student’s t distributions A family of distributions identified by “Student” (William Sealy Gosset) in 1908 t family members are identified by their degrees of freedom, df. t distributions are similar to z distributions but with broader tails As df increases → t tails get skinnier → t become more like z

t table (Table C) Table C: Entries  t values Rows  df Columns  probabilities Use Table C to look up t values and probabilities

Left tail: Pr(T 9 < ) = 0.10 Right tail: Pr(T 9 > 1.383) = 0.10 Understanding Table C Let t df,p ≡ a t value with df degrees of freedom and cumulative probability p. For example, t 9,.90 = Table C. Traditional t table Cumulative p Upper-tail p df =

§11.3 One-Sample t Test A. Hypotheses. H 0 : µ = µ 0 vs. H a : µ ≠ µ 0 (two-sided) [ H a : µ µ 0 (right-sided)] B. Test statistic. C. P-value. Convert t stat to P-value [table C or software]. Small P  strong evidence against H 0 D. Significance level (optional). See Ch 9 for guidelines.

One-Sample t Test: Example Statement of the problem: Do SIDS babies have lower than average birth weights? We know from prior research that the mean birth weight of the non-SIDs babies in this population is 3300 grams We study n = 10 SIDS babies, determine their birth weights, and calculate x-bar = and s = 720. Do these data provide significant evidence that SIDs babies have different birth weights than the rest of the population?

A.H 0 : µ = 3300 versus H a : µ ≠ 3300 (two-sided) B. Test statistic C. P = [next slide] Weak evidence against H 0 D. (optional) Data are not significant at α =.10 One-Sample t Test: Example

Converting the t stat to a P-value t stat  P-value via Table C. Wedge |t stat | between critical value landmarks on Table C. One-tailed 0.05 < P < 0.10 and two-tailed 0.10 < P < t stat  P-value via software. Use a software utility to determine that a t of −1.80 with 9 df has two- tails of Table C. Traditional t table Cumulative p Upper-tail p df = |t stat | = 1.80

Two-sided P-value associated with a t statistic of and 9 df

§11.4 Confidence Interval for µ Typical point “estimate ± margin of error” formula t n-1,1-α/2 is from t table (see bottom row for conf. level) Similar to z procedure except uses s instead of σ Similar to z procedure except uses t instead of z Alternative formula:

Confidence Interval: Example 1 Let us calculate a 95% confidence interval for μ for the birth weight of SIDS babies.

Confidence Interval: Example 2 Data are “% of ideal body weight” in 18 diabetics: {107, 119, 99, 114, 120, 104, 88, 114, 124, 116, 101, 121, 152, 100, 125, 114, 95, 117}. Based on these data we calculate a 95% CI for μ.

§11.5 Paired Samples Paired samples: Each point in one sample is matched to a unique point in the other sample Pairs be achieved via sequential samples within individuals (e.g., pre-test/post-test), cross-over trials, and match procedures Also called “matched-pairs” and “dependent samples”

Example: Paired Samples A study addresses whether oat bran reduce LDL cholesterol with a cross-over design. Subjects “cross-over” from a cornflake diet to an oat bran diet. –Half subjects start on CORNFLK, half on OATBRAN –Two weeks on diet 1 –Measures LDL cholesterol –Washout period –Switch diet –Two weeks on diet 2 –Measures LDL cholesterol

Example, Data Subject CORNFLK OATBRAN

Calculate Difference Variable “DELTA” Step 1 is to create difference variable “DELTA” Let DELTA = CORNFLK - OATBRAN Order of subtraction does not materially effect results (but but does change sign of differences) Here are the first three observations: Positive values represent lower LDL on oatbran ID CORNFLK OATBRAN DELTA ↓ ↓ ↓ ↓

Explore DELTA Values Stemplot |-0|42 |+0|0133 |+0| ×1 Here are all the twelve paired differences (DELTAs): 0.77, 0.85, − 0.45, −0.26, 0.30, 0.86, 0.60, 0.62, 0.31, 0.72, 0.09, 0.16 EDA shows a slight negative skew, a median of about 0.45, with results varying from −0.4 to 0.8.

Descriptive stats for DELTA Data (DELTAs): 0.77, 0.85, − 0.45, −0.26, 0.30, 0.86, 0.60, 0.62, 0.31, 0.72, 0.09, 0.16 The subscript d will be used to denote statistics for difference variable DELTA

95% Confidence Interval for µ d A t procedure directed toward the DELTA variable calculates the confidence interval for the mean difference. “Oat bran” data:

Paired t Test Similar to one-sample t test μ 0 is usually set to 0, representing “no mean difference”, i.e., H 0 : μ = 0 Test statistic:

Paired t Test: Example “Oat bran” data A. Hypotheses. H 0 : µ d = 0 vs. H a : µ d  0 B. Test statistic. C. P-value. P = (via computer). The evidence against H 0 is statistically significant. D. Significance level (optional). The evidence against H 0 is significant at α =.05 but is not significant at α =.01

SPSS Output: Oat Bran data

§11.6 Conditions for Inference t procedures require these conditions: SRS (individual observations or DELTAs) Valid information (no information bias) Normal population or large sample (central limit theorem)

The Normality Condition The Normality condition applies to the sampling distribution of the mean, not the population. Therefore, it is OK to use t procedures when: The population is Normal Population is not Normal but is symmetrical and n is at least 5 to 10 The population is skewed and the n is at least 30 to 100 (depending on the extent of the skew)

Can a t procedures be used? Dataset A is skewed and small: avoid t procedures Dataset B has a mild skew and is moderate in size: use t procedures Data set C is highly skewed and is small: avoid t procedure

§11.7 Sample Size and Power Questions: –How big a sample is needed to limit the margin of error to m? –How big a sample is needed to test H 0 with 1−β power at significance level α? –What is the power of a test given certain conditions? In this presentation, we cover only the last question

Power where: α ≡ (two-sided) alpha level of the test Δ ≡ “the mean difference worth detecting” (i.e., the mean under the alternative hypothesis minus the mean under the null hypothesis) n ≡ sample size σ ≡ standard deviation in the population Φ(z) ≡ the cumulative probability of z on a Standard Normal distribution [Table B] with.

Power: Illustrative Example SIDS birth weight example. Consider the SIDS illustration in which n = 10 and σ is assumed to be 720 gms. Let α = 0.05 (two-sided). What is the power of a test under these conditions to detect a mean difference of 300 gms?