Pp # 5 CHAPT 7 Hypothesis Testing Applied to Means 1. Single Sample or One Sample t-Test AKA student t-test. 2. Two Independent sample t-Test, AKA Between Subject Designs or Matched subjects Experiment. 3. Related Samples t-test or Repeated Measures Experiment AKA Within Subject Designs or Paired Sample t-Test .

Degrees of Freedom df=n-1

Assumption of the t-test (Parametric Tests) 1.The Values in the sample must consist of independent observations. 2. The population sample must be normal. 3. Use a large sample n ≥ 30

1.The Values in the sample must consist of independent observations.

2. The population sample must be normal.

Standard Error (σM) The standard Deviation of the distribution of sample means (σM) is called Standard Error of M (σM). The standard error provides a measure of how much distance is expected on average between sample means (M) and the population mean (µ). σM= σ/√n

The estimated Standard Error (SM) The estimated Standard Error (SM) is used as an estimate of the real Standard Error of M (σM) when the value of σ is unknown. It is computed using the sample variance or sample standard deviation and provides an estimate of the standard distance between a sample means (M), and the population mean (µ). SM= S/√n or SM= √S²/n

The estimated Standard Error (SM) Z=(X-µ)/σM or Z= Obtained difference between data and hypothesis/Standard Distance between M and µ.

Calculations for t-test  

FYI HYPOTHESIS for Z

FYI Steps in Hypothesis-Testing Step 1: State The Hypotheses H0 : µ ≤ 100 average H1 : µ > 100 average Statistics: Because the Population mean or µ is known the statistic of choice is z-Score

FYI Hypothesis Testing Step 2: Locate the Critical Region(s) or Set the Criteria for a Decision

FYI Directional Hypothesis Test

FYI None-directional Hypothesis Test

FYI Hypothesis Testing Step 3: Computations/ Calculations or Collect Data and Compute Sample Statistics Z Score for Research   ZZZ 

FYI Hypothesis Testing Step 3: Computations/ Calculations or Collect Data and Compute Sample Statistics    

Calculations for t-test Step 3: Computations/ Calculations or Collect Data and Compute Sample Statistics  

Hypothesis Testing Step 4: Make a Decision

Inferential Statistics t-Static Single Sample or One Sample t-Test t-test is used to test hypothesis about an unknown population mean (µ) when the value of σ or σ² is unknown (we estimate them by using S or S²). Ex. Is this year class know more about STATS than the last year? Mean for the last year class µ=85 Mean for this year class M=88 Note: We don’t know what the average/mean STATS score should be for the population. We only compare this year scores with the last year. ****Sample data can only be considered as estimates of population values.**** Inferential Statistics t-Static Single Sample or One Sample t-Test

FYI Variability SS, Standard Deviations and Variances X σ² = ss/N Pop 1 σ = √ss/N 2 4 s = √ss/df 5 s² = ss/n-1 or ss/df Sample SS=Σx²- (Σx)²/N SS=Σ( x-μ)² Sum of Squared Deviation from Mean

FYI d=Effect Size for Z For t-test Use S instead of σ (next slide)  

Cohen's d and Effect Size d=Effect Size/Cohn’ d Cohen's d and Effect Size A statistically significant result does not mean that the result is practically significant (treatment effect). To correct this problem, it is recommended that whenever researchers report a statistically significant effect, they also provide a report of the effect size (see the guidelines presented by L. Wilkinson and the APA Task Force on Statistical Inferences, 1999). Therefore for different hypothesis test (test statistics) there are different options for measuring and reporting effect size.

Use Phi Coefficient and Cramer,s V for Chi Square d=Effect Size/Cohn’ d The “effect size” gives an indication of whether something is practically significant. There are different ways of calculating an effect size. r which is the correlation coefficient or R² which is the coefficient of determination Use Eta squared ή² for ANOVA Use Cohen’s d for t-test Use Phi Coefficient and Cramer,s V for Chi Square

Evaluating effect size with Cohen’s d Magnitude of d Evaluation of effect size d= 0.2 Small effect (mean difference around 0.20 standard deviation) d= 0.5 Medium effect (mean difference around 0.5 standard deviation) d= 0.8 Large effect (mean difference around 0.8 standard deviation)

percentage of Variance accounted for by the Treatment Percentage of Variance Explained r²=0.01-------- Small Effect r²=0.09-------- Medium Effect r²=0.25-------- Large Effect

Cohn’s d=Effect Size for Single Sample t Use S instead of σ for t-test d = (M - µ) S S= (M - µ) d M= (d . s) + µ µ= (M – d) s

Percentage of Variance Accounted for by the Treatment (similar to Cohen’s d) Also known as ω² Omega Squared (power of a test)

Problem 1 A supervisor has prepared an “Optimism Test” that is administered yearly to factory employees. The test measures how each employee feels about its future. The higher the score, the more optimistic the employee. Last year’s employees had a mean score of μ=15. A sample of n=9 employees from this year was selected and tested..

Problem 1 The scores for these employees are 7, 12, 11, 15, 7, 8, 15, 9, and 6, which produced a sample mean of M=10 with SS=94. On the basis of this sample, can the supervisor conclude that this year’s employees has a different level of optimism? Alpha =0.05, two tailed Note that this hypothesis test will use a t-statistic because the population variance σ² is not known (S²). USE SPSS Set the level of significance at α=.05 for two tails

Null Hypothesis t-Statistic: If the Population mean or µ and the sigma are unknown the statistic of choice will be t-Static 1. Single (one) Sample t-statistic (test) Step 1 H0 : µ optimism = 15 H1 : µ optimism ≠ 15

Step 2 Locate the Critical regions

Calculations for t-test Step 3: Computations/ Calculations or Collect Data and Compute Sample Statistics M-μ t= s Sm Sm= or √n df=n-1 s² = SS/df Sm= estimated standard error of the mean

Step 4 Make a Decision Step 2

Bootstrap Bootstrapping is a method for deriving robust estimates of standard errors and confidence intervals for estimates such as the mean, median, proportion, odds ratio, correlation coefficient or regression coefficient.

Problem 2 Infants, even newborns prefer to look at attractive faces (Slater, et al., 1998). In the study, infants from 1 to 6 days old were shown two photographs of women’s face. Previously, a group of adults had rated one of the faces as significantly more attractive than the other. The babies were positioned in front of a screen on which the photographs were presented. The pair of faces remained on the screen until the baby accumulated a total of 20 seconds of looking at one or the other. The number of seconds looking at the attractive face was recorded for each infant.

Problem 2 Attractive 10 11 16 18 13 17 M=13 S=3

Problem 2 Suppose that the study used a sample of n=9 infants and the data produced an average of M=13 seconds for attractive face with S=3. Set the level of significance at α=0.01 and then 0.05 for two tailed Note that all the available information comes from the sample. Specifically, we do not know the population mean μ or the population standard deviation σ. On the basis of this sample, can we conclude that infants prefer to look at attractive faces?

Null Hypothesis t-Statistic: If the Population mean or µ and the sigma are unknown the statistic of choice will be t-Static 1. Single (one) Sample t-statistic (test) Step 1 H0 : µ attractive = 10 seconds H1 : µ attractive ≠ 10 seconds

STEP 2None-directional Hypothesis Test Critical value of t=2.306

Calculations for t-test Step 3: Computations/ Calculations or Collect Data and Compute Sample Statistics  

Hypothesis Testing Step 4: Make a Decision

t-Stats 1. Single Sample or One Sample t-Test AKA student t-test. 2. Two Independent sample t-Test, AKA Between Subject Designs or Matched subjects Experiment. 3. Related Samples t-test or Repeated Measures Experiment AKA Within Subject Designs or Paired Sample t-Test .

2. Two Independent sample t-Test AKA Between Subject Designs or Matched subjects Experiment

t-test ANOVA

Two Independent Sample t-test An independent measures study uses a separate group of participants (samples) to represent each of the populations or treatment conditions being compared.

Two Independent Sample t-test Null Hypothesis: If the Population mean or µ is unknown the statistic of choice will be t-Static Two independent sample t-test, Matched-Subject Experiment, or Between Subject Design Step 1 H0 : µ1 -µ2 = 0 H1 : µ1 -µ2 ≠ 0

None-directional Hypothesis Test Step 2

STEP 3  

Estimated Standard Error S(M1-M2) The estimated standard error measures how much difference is expected, on average, between a sample mean difference and the population mean difference. In a hypothesis test, µ1 - µ2 is set to zero and the standard error measures how much difference is expected between the two sample means.

Estimated Standard Error S(M1-M2)=

(1) Pooled Variance

(2) Pooled Variance

Step 4

Measuring d=Effect Size for the independent measures

Estimated d

Estimated d

Percentage of Variance Accounted for by the Treatment (similar to Cohen’s d) Also known as ω² Omega Squared

Problem 1 Research results suggest a relationship Between the TV viewing habits of 5-year-old children and their future performance in high school. For example, Anderson, Huston, Wright & Collins (1998) report that high school students who regularly watched Sesame Street as children had better grades in high school than their peers who did not watch Sesame Street.

Problem 1 The researcher intends to examine this phenomenon using a sample of 20 high school students. She first surveys the students’ s parents to obtain information on the family’s TV viewing habits during the time that the students were 5 years old. Based on the survey results, the researcher selects a sample of n1=10

Use non-directional or two-tailed test Problem 1 students with a history of watching “Sesame Street“ and a sample of n2=10 students who did not watch the program. The average high school grade is recorded for each student and the data are as follows: Set the level of significance at α=.05 and Use non-directional or two-tailed test

Problem 1 Average High School Grade Watched Sesame St (1). Did not Watch Sesame St.(2) 86 90 87 89 91 82 97 83 98 85 99 79 94 86 89 81 92 92 n1=10 n2=10 M1=93 M2= 85 SS1=200 SS2=160

Two Independent Sample t-test Null Hypothesis: Two Independent Sample t-test, Matched-Subject Experiment, or Between Subject Design Step 1. H0 : µ1 -µ2 = 0 H1 : µ1 -µ2 ≠ 0

Two Independent Sample t-test Null Hypothesis: Two independent sample t-test, Matched-Subject Experiment, or Between Subject Design  directional or one-tailed test Step 1. H0 : µ Sesame St . ≤ µ No Sesame St. H1 : µ Sesame St. > µ No Sesame St.

Step 2

STEP 3  

Estimated Standard Error S(M1-M2)=

(1) Pooled Variance s² P

Estimated Standard Error S(M1-M2)=

Step 4

Measuring d=Effect Size for the independent measures

FYI We use the Point-Biserial Correlation (r) when one of our variable is dichotomous, in this case (1) watched Sesame St. and (2) didn’t watch Sesame St.

Problem 2 In recent years, psychologists have demonstrated repeatedly that using mental images can greatly improve memory. Here we present a hypothetical experiment designed to examine this phenomenon. The psychologist first prepares a list of 40 pairs of nouns (for example, dog/bicycle, grass/door, lamp/piano). Next, two groups of participants are obtained (two separate samples). Participants in one group are given the list for 5 minutes and instructed to memorize the 40 noun pairs.

Problem 2 Participants in another group receive the same list of words, but in addition to the regular instruction, they are told to form a mental image for each pair of nouns (imagine a dog riding a bicycle, for example). Later each group is given a memory test in which they are given the first word from each pair and asked to recall the second word. The psychologist records the number of words correctly recalled for each individual. The data from this experiment are as follows: Set the level of significance at α=.01 for two tailed test.

Problem 2 Data (Number of words recalled) Group 1 (Images) Group 2 (No Images) 19 23 20 22 24 15 30 16 31 18 32 12 27 19 22 14 25 18 26 18 26 25 n1=12 n2=12 M1=26 M2= 18 SS1=200 SS2=160

Two Independent Sample t-test Null Hypothesis: Step 1 H0 : µ1 -µ2 = 0 H1 : µ1 -µ2 ≠ 0

Step 2

STEP 3  

(1) Pooled Variance s² P

Estimated Standard Error S(M1-M2)=

Step 4