conceptual understanding

Conceptual Understanding

Source SS df MS F

Between -- -- -- --

Within 152 -- - -

Total 182 --

Complete the above table for an ANOVA having 3 levels of the independent variable and n = 20. Test for significant at .05.


Source SS df MS F

Between 30 2 15 5.62*

Within 152 57 2.67

Total 182 59

Fcrit = 3.18

Complete the above table for an ANOVA having 3 levels of the independent variable and n = 20. Test for significant at .05.

Fcrit (2, 57) = 3.15


• Under what circumstance will the F ratio, over the long run, approach 1.00? Under what circumstances will the F ratio be greater than 1.00?


• Under what circumstance will the F ratio, over the long run, approach 1.00? Under what circumstances will the F ratio be greater than 1.00?

• F ratio will approach 1.00 when the null hypothesis is true

• F ratio will be greater than 1.00 when the null hypothesis is not true


A B C

3 5 7

3 5 7

3 5 7

3 5 7

Without computing the SS within, what must its value be? Why?


A B C

3 5 7

3 5 7

3 5 7

3 5 7

The SS within is 0. All the scores within a group are the same (i.e., there is NO variability within groups)

Practice

• You are interested in comparing the taste preference of four sodas. Subjects drank one of four sodas and asked to rank it on a 1-10 scale of how much they liked the taste.

Results

Pepsi Coke RC MountainDew

8 6 8 9

7 8 6 5

9 7 8 8

9 9 7 7

Practice

• Is there a significant (.01) difference in how much these sodas were liked?

Step 1: State the Hypothesis

• H0: P = C = R =M

• H1: The four population means are not all equal

Step 2: Find F-Critical

• Step 2.1• Need to first find dfbetween and dfwithin

• Dfbetween = 3 (k = number of groups)

• dfwithin = 12 (N = total number of observations)

• dftotal = 15

• Check yourself• 15 = 3 + 12

Step 2: Find F-Critical

• Step 2.2

• Look up F-critical using table F on pages 370 - 373.

• F (3,12) = 5.95

Step 3.1: Ingredients

X = 121X2 = 937Tj

2 = 3671

• N = 16

• n = 4

Step 3.2: Calculate SS

X = 121

X2 = 937

Tj2

= 3671

N = 16

n = 4• SStotal


• SStotal

937121

1621.94

X = 121

X2 = 937

Tj2

= 3671

N = 16

n = 4


• SSWithin

X = 121

X2 = 937

Tj2

= 3671

N = 16

n = 4


• SSWithin

9373671

419.25

X = 121

X2 = 937

Tj2

= 3671

N = 16

n = 4


• SSBetween

X = 121

X2 = 937

Tj2

= 3671

N = 16

n = 4


• SSBetween

3671

4

121

16

2.69

X = 121

X2 = 937

Tj2

= 3671

N = 16

n = 4


• Check!

• SStotal = SSBetween + SSWithin


• Check!

• 21.94 = 2.69+ 19.25

Step 3.3: Calculate MS


2.69

3.90

Calculating this Variance Ratio


19.25

121.60

Step 3.4: Calculate the F value

.90

1.60.56

Step 3.4: Calculate the F value

Step 4: Decision

• If F value > than F critical– Reject H0, and accept H1

• If F value < or = to F critical– Fail to reject H0

Step 4: Decision

• If F value > than F critical– Reject H0, and accept H1

• If F value < or = to F critical– Fail to reject H0

F value = .56

F crit = 5.95

Step 5: Create the Summary Table

Source SS df MS F

Between 2.69 3 .90 .56

Within 19.25 12 1.60

Total 21.94 15

Step 6: Put answer into words• Question: Is there a significant (.01) difference

in how much these sodas were liked?

• H0: P = C = R =M

• There was not a significant difference in how much these sodas were liked. On average, they were all liked equally as well.

Practice

• Odometers measure automobile mileage. Suppose 12 cars drove exactly 10 miles and the following mileage figures were recorded. Determine if, on average, the odometers were accurate (Alpha = .05).

9.8, 10.1, 10.3, 10.2, 9.9, 10.4, 10.0, 9.9, 10.3, 10.0, 10.1, 10.2

One-sample t-test

• H1 = Mean not equal to 10• H0 = Mean = 10

• t critical (11) = 2.201• t obs = 1.86

• The odometers did not record a siginficantly different distance than what was driven.

Practice

• A research study was conducted to examine the differences between older and younger adults on perceived life satisfaction. A pilot study was conducted to examine this hypothesis. Ten older adults (over the age of 70) and ten younger adults (between 20 and 30) were given a life satisfaction test with high scores indicative of high life satisfaction. Determine if age has an affect (.05) on life satisfaction.

Older Adults Younger Adults

Mean = 44.5 Mean = 28.1

S = 8.68 S = 8.54

• tobs = 4.257

• tcrit = 2.101

• Age does seem to “affect” life satisfaction.

Review

• Compare one sample to another value

– One sample t-test

• Compare two independent samples to each other

– Two sample independent t-test (equal or unequal n)

• Compare two dependent samples to each other

– Two sample dependent t-test

• Compare two or more independent samples to each other

– One-way ANOVA

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