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Analysis of variance (ANOVA) - Hypothesis Testing

Analysis of variance (ANOVA)

Lessons

Notes:
Recall:
σ2=(x1μ)2+(x2μ)2+(xnμ)2n\sigma^2=\frac{(x_1-\mu)^2+(x_2-\mu)^2+ \cdots (x_n-\mu)^2}{n}

x=x1+x2++xnn\overline{x}=\frac{x_1+x_2+ \cdots +x_n}{n}

Degrees of Freedom
The degrees of freedom for a calculation is the number of variables that are free to vary. Think of calculating the mean of several variables.

d.f.=n1d.f.=n-1

Sums of Squares:

The Sum of Squares Within Groups (SSW) is calculated by first finding the sum of squares for each individual group, and then adding them together.

The Sum of Squares Between Groups (SSB) is calculated by first finding the mean for all the groups (the Grand Mean) and then seeing what is the sum of squares from each individual group to the Grand Mean.

The Total Sum of Squares (TSS or SST) is just the sum of sqaures of the every single item from all the groups. Just imagine that all the groups come together to form one big group.

Total Sum of Squares = Sum of Squares Within + Sum of Squares Between (TSS=SSW+SSB)

Hypothesis Testing with F-Distribution
This method is just the test that the variances between groups do not vary.

F=betweengroupvariabilitywithingroupvariabilityF=\frac{between\;group\;variability}{within\;group\;variability} =SSBdfSSWdf=\frac{\frac{SSB}{df}}{\frac{SSW}{df}}

F(dfSSB,dfSSW)F(df_{SSB},df_{SSW}) is the critical value for an F-distribution
  • 1.
  • 2.
    Determining Degrees of Freedom
    A test was done to study the reaction time of car drivers at different periods of the day

    Reaction Time of Drivers (seconds)

    Morning:

    Afternoon:

    Evening:

    Night:

    1.32

    0.25

    2.34

    1.54

    0.71

    1.48

    1.75

    1.98

    2.27

    1.24

    0.64

    1.76

    0.57

    0.89

    0.98

    1.15

    x=1.2175\overline{x}=1.2175

    x=0.965\overline{x}=0.965

    x=1.4275\overline{x}=1.4275

    x=1.6075\overline{x}=1.6075

    x=1.304375\overline{x}=1.304375

  • 3.
    Determining the Sum of Squares
    The following case study was done on what type of beverages office workers drink in the morning and their productivity.

    Juice/Milk Drinkers:

    Tea Drinkers:

    Coffee Drinkers:

    3

    5

    8

    5

    5

    6

    3

    6

    7

    1

    4

    7

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Analysis of variance (ANOVA)

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