Hypothesis Testing with t Tests

Hypothesis Testing with t Tests A l Cl Arlo Clark-Foos kF

Using Samples to Estimate Population Parameters Acknowledge g error y Smaller samples, less spread y

Σ( X − M ) s= N −1

2

The t Statistic y

Indicates the distance of a sample p mean from a population mean in terms of the standard error s = s M

N

Hypothesis Tests: Single--Sample t Tests Single

“Guinness is the best beer available, it does not need d advertising d as its quality l willll sellll it, and those who do not drink it are to be sympathized with rather than advertised to. to”

Hypothesis Tests: Single--Sample t Tests Single y

Hypothesis test in which we compare data from one sample to a population for which we know the mean but not the standard deviation.

y

Degrees of Freedom: ◦ The number of scores that are free to vary when estimating a population parameter from a sample ◦ df = N – 1 (for a Single-Sample t Test)

One Tailed vs. vs Two Tailed Tests

Six Steps for Hypothesis Testing 1. Identifyy 2. State the hypotheses 3 Characteristics of the comparison 3. distribution 4 Critical values 4. 5. Calculate 6. Decide

Single--Sample t Test: Example Single Participation p in therapy py sessions y Contract to attend 10 sessions y μ = 4.6 46 y

Sample: y 6, 6, 12, 7, 8 y

Single--Sample t Test: Example Single Identifyy

1.

◦ ◦

Pop 1: All clients who sign contract g contract Popp 2: All clients who do not sign

◦

Distribution of means

◦

Test: Population mean is known but not standard deviation Æ single-sample t test x

Assumptions

Single--Sample t Test: Example Single State the null and research hypotheses yp

2.

◦

H0: Clients who sign the contract will attend the same number of sessions as those who do not sign the contract. x

◦

μ1 = μ2

H1: Clients who sign the contact will attend a different number of sessions than those who do not sign the contract. contract x

μ1 ≠ μ2

Single--Sample t Test: Example Single Determine characteristics of comparison distribution 2 μM = μ = 4.6, Σ( X − M ) s= = 2.490 sM = 1.114 N −1 3.

s 2.490 sM = = = 1.114 N 5

Single--Sample t Test: Example Single Determine critical values or cutoffs

4.

◦ ◦

df = N-1 = 5-1 = 4, p = .05 t = ± 2.776

Single--Sample t Test: Example Single Calculate the test statistic

5.

( M − μ M ) (7.8 − 4.6) t= = = 2.873 sM 1.114

M k a decision Make d ii

6 6.

◦ ◦

2.873 > 2.776, we reject the null Clients who sign a contract will attend more sessions than those who do not sign a contract. contract

Reporting Results in APA Format 1. 2. 3. 4.

Write the symbol for the test statistic (e.g., z or t) Write the degrees of freedom in parentheses Write an equal sign and then the value of the test statistic (2 decimal places) Write a comma and then whether the p value associated with the test statistic was less than or greater than the cutoff p value of .05 05 t(4) = 22.87, 87 p < .05 05

Hypothesis tests for two samples y

Conditions must be equivalent q (controlled) for a fair test

y

Beer Tasting: Between-groups vs. Withingroups

Paired--Samples t Test Paired y

Used to compare p 2 means ffor a withingroups design, a situation in which every pparticipant p is in both samples p (paired/dependent)

y

Difference Scores: X1 – Y1, X2 – Y2, …

Paired--Samples t Test Paired Identifyy 2. State null & research hyp. 3 Determine characteristics of 3. comparison distribution H 0 : μ1 = μ2 1.

Paired--Samples t Test Paired Determine cutoffs 5. Calculate test statistic (M − μM ) 4.

t=

6 6.

M k a decision Make d ii

sM

Independent Samples t Test y

Used to compare p 2 means ffor a betweengroups design, a situation in which each pparticipant p is assigned g to onlyy one condition.


What p percentage g of cartoons do men and woman consider funny?

Women: y Men: y

84 97 58 90 88 90 52 97

86

Independent Samples t Test Identifyy

1.

◦

Population 1: Women exposed to humorous cartoons Population 2: Men exposed to humorous cartoons Distribution: Differences between means

◦ x

◦

Not mean differences

Assumptions are the same x

We meet one here: DV is interval/scale

Independent Samples t Test State null and research hypotheses

2. ◦ ◦

◦ ◦

H 0 : μ1 = μ2 Women will categorize the same number of cartoons as funny as will men. men H 1 : μ1 ≠ μ2 Women Wo e will w categorize catego e a different number of cartoons funny as as will men.

Independent Samples t Test Determine characteristics of comparison distribution

3.

◦ ◦

H0: μ1 = μ2 Measure of spread: a) Calculate variance for each sample b) Pool variances, variances accounting for sample size c) Convert from squared standard deviation to squared standard error d) Add the two variances e) Take square root to get estimated standard error for distribution of differences between means.

Independent Samples t Test a) Calculate variance for each sample 2 Σ ( X − M ) 868.752 s X2 = = = 289.584 N −1 4 −1

2 Σ ( Y − M ) 1314 4 1314.4 sY2 = = = 328.6 N −1 5 −1


Degrees g of Freedom dfX = N – 1 = 4 – 1 = 3 dfY = N – 1 = 5 – 1 = 4 dfTotal = dfX + dfY = 3 + 4 = 7

Pooled Variance b)

y

Pool variances, accounting for sample size

Weighted average of the two estimates of variance – one from each sample – that are calculated when conducting d i an iindependent d p d samples pl t test. s

2 Pooled

s

2 Pooled

⎛ df X =⎜ ⎝ dfTotal

⎞ 2 ⎛ dfY ⎟ sX + ⎜ ⎠ ⎝ dfTotal

⎞ 2 ⎟ sY ⎠

⎛3⎞ ⎛4⎞ = ⎜ ⎟ 289.584 + ⎜ ⎟ 328.6 ⎝7⎠ ⎝7⎠

2 sPooled = 124.107 + 187.771 = 311.878

Independent Samples t Test c)) Convert from squared q standard deviation to squared standard error 2 sPooled = 311.878 s

2 MX

2 sPooled 311 878 311.878 = = = 77.970 N 4

s

2 MY

2 sPooled 311 878 311.878 = = = 62.376 N 5

d) Add the two variances 2 sDifference = sM2 X + sM2 Y = 77.970 + 62.376 = 140.346

e) Take square root to get estimated standard error for distribution of differences between means. 2 sDifference = sDifference = 140.346 = 11.847

Independent Samples t Test Determine critical values

4.

◦ 5.

dfTotal = 7

p = .05

t = ± 2.365

Calculate a test statistic ⎡⎣( M X − M Y ) − ( μ X − μY ) ⎤⎦ ( M X − M Y ) t= = sDifference sDifference 82.25 − 82.6 ) ( t= = −.03 11 847 11.847

Independent Samples t Test 6.

Make a decision

y

Fail to reject null hypothesis ◦

Men and women find cartoons equally humorous