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Comparing Sample Means

Lecture Goals

You should:

- understand how to use and calculate t scores to compare related and independent means

- be able to test hypotheses for related & independent means

- understand the advantages of using different designs for comparing means

Reading

Chapter 14

Comparing 2 Means

- Which teaching technique is better?

- Do men and women's attitudes differ?

- Are people better after treatment?

- Do people work better after training?

- Do people's brains grow with age?

Related Samples

- Is same person better after treatment

- Is same person better after training

Independent Samples

- Which drug helps more A or B?

- Who scored higher group A or B?

Related Samples

- Usually within subjects manipulations

- Between subjects manipulations on matched samples

Independent Samples

- Between subjects manipulations

Does our new the diet plan work?

How to answer with Related Samples:

Repeated Measures Design = measure same people before and after

Matched Subjects Design = simulated repeated measures deign

= have a 2 groups & match subjects on characteristics (age, gender, SES)

Person Before After
Weight Weight

    A             255         251
    B             234         228
    C             219         207
    D             188         179
    E             154         150
    F             201         193
    G             188         187
    H             167         155

Comparing 2 Related Means

Research Question = about difference between: 1) time 1 (before) and time 2 (after)
or 2) paired subjects

Difference scores = difference in scores between paired scores Person Before After Difference
Weight Weight

    A             255     -     251         = 4
    B             234     -     228         = 6
    C             219     -     207         = 12
    D             188     -     179         = 9
    E             154     -     150         = 4
    F             201     -     193         = 8
    G             188     -     189         = -1
    H             167     -     155         = 12

Before:

t = sample mean-m= Observed Value - Expected Value
s_mean Estimated Standard Deviation

Now - interested in the difference scores

Observed Difference = D

Expected Difference if H_o = 0

t = Observed Value - Expected Value = D - 0
Estimated Standard Deviation Estimated S. D.

Now - interested in the difference scores

Estimated S.D. = standard error of the difference scores

t = Observed Value - Expected Value = D - 0
Estimated Standard Deviation s_D-bars

Before:

s_means= s/ n

see formula

s_D-bars_{=
sDif. scores / n}

see two formulae

See t-formula

What do we need?

See symbols

Person Before After Difference
Weight Weight

N = 8 S D_i = 54
D = 54/8 = 6.75

Person Before After Diff. Diff.²

    A             255         251     4         16
    B             234         228     6         36
    C             219         207     12     144
    D             188         179     9         81
    E             154         150     4         16
    F             201         193     8         64
    G             188         189     -1         1
    H             167         155    12     144

S D_i²= 502 What do we need?

D = 6.75
S D_i = 54
S D_i² = 502
N = 8 See t-formula worked out

Alternative 7 Step Process for Testing 2 Related Means

State null hypothesis

State alternative hypothesis

Set level of significance

Determine critical t-score & region

Calculate the t-score

Compare to make decision about H_o

Draw a conclusion

A researcher wants to know if a new diet plan helps people to lose weight. a = .05. The data are below.

Person Before After
Weight Weight

Step 1: State the Null Hypothesis

H_{o :}_mbefore £m after (m _D_£0) Step 2: State the Alternative Hypothesis H_{A :}_mbefore > m after (m _D> 0) Step 3: Set Level of Significance

a = .05

Step 4: Determine critical t-score & region

t for df = 7 & a = .05_1-tailed = 1.895

see graph

Step 5: Calculate t-score

see t-formula worked out

Step 6: Decision About H_o

t = 2.749 = in critical region

see graph

Step 6: Decision About H_o

If |t_calc| ³ |t_crit| then Reject H_o

If |t_calc| < |t_crit| then Fail to Reject H_o

Here: 2.749 > 1.895 so Reject H_o

Step 7: Draw a Conclusion

The data show that the diet plan helped people lose weight

Does our new the diet plan work?

How to answer with Independent Samples:

Experimental-Control Group Design = compare people who get manipulation to those who don't

= can compare more than one experimental group

= measure all at same time

No Plan Group Diet Plan Group

Person     Weight    Person     Weight
    A             255             L             251
    B             234             M             228
    C             219             N             207
    D             188             O             179
    E             154             P             150
    F             201             Q             193
    G             188             R             187
    H             167             S             155

Comparing 2 Independent Means

Research Question = about difference between the means of each group

Now we are comparing group means, NOT the change in each subject or across matched subjects

Now - interested in the difference between the means of each group

_ _
Observed Difference = X₁ - X₂

Expected Difference if H_o = 0

t = Observed Value - Expected Value = (X₁ - X₂) - 0
Estimated Standard Deviation Estimated S. D.

Now - interested in the difference between the means of each group

Estimated S.D. = standard error of the difference between the means
= Observed Value - Expected Value = (X₁ - X₂) - 0
Estimated Standard Deviation s_{mean1 - mean2}

Estimated standard error

Now idea = weighted average of sample variances

s_{mean1 - mean2} = SS₁ + SS₂
df₁ + df₂

Formula for estimated standard error

See formula To average we divide by n of both samples

See formula

t = Observed Value - Expected Value = (X₁ - X₂) - 0
Estimated Standard Deviation s_{mean1 - mean2}

See formula

What do we need to solve for t ?

X₁ = ? SS₁= ?

X₂ = ? SS₁= ?

N₁ = ? S X_i = ?

N₂ = ? S X₂ = ?

S X₁² = ?

S X₂² = ?

N₁ = 8 N₂ = 8

S X_i = 1606 S X₂ = 1550

No Plan Group Diet Plan Group

Weight Weight² Weight Weight²

    255         65025         251         63001
    234         54756         228         51984
    219         47961         207         42849
    188         35344         179         32041
    154         23716         150         22500
    201         40401         193         37249
    188         35344         187         34969
    167         27889         155         24025

SX_i² = 330446 S X₂² = 308618

What do we need to solve for t ?

X₁ = 1606/8 = 200.75 S X_i = 1606

X₂ = 1550/8 = 193.75 S X₂ = 1550

N₁ = 8

N₂ = 8

S X₁² = 330446

S X₂² = 308618

SS₁ = S X₁² - (S X₁)²
N₁

= 330446 - (1606)²
8

= 8041.5

SS₂ = S X₂² - (S X₂)²
N₂

= 308618 - (1550)²
8

= 8305.5

see t-formula worked out

Assumptions: Comparing Ind. Means

H_o = both samples from same pop....so

    1) X1 and X2 from normal distribution

    2) X1 - X2's normally distributed

3) Homogeneity of variance (s₁ = s ₂)

t tests = robust = still work when assumptions are NOT met when n ³ 30

Within s vs. Between s Designs

Within Subjects Design

Advantages

Compare people to themselves

No individual diffs. across groups - this can reduce variance and increase p(reject H_o)

Disadvantages

Order effects

Practice effects

Fatigue effects

Motivation effects

** Must counterbalance **
Between Subjects Design

Eliminates _____ across conditions

Order effects

Practice effects

Fatigue effects

Motivation effects

Compares people to other people

Need random selection and assignment to reduce individual differences across grps. -
Individual differrences can increase variance and decrease p(reject H_o)

Homework

Chapter 14

Problems 9, 10, 12, 13, 16a only, 20

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