BUSOBA 2320: W8 Rec Notes AND Videos

REC

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H0:

μ1 = μ2 = … = μk

Ha:_ , or HA:

• Not all μi are the same

• At least two μi are different

Factor =

• categorical sorting variable that creates/defines the different populations to compare

• VID: categorical variable with 2 or more levels that sort the response variable values into groups and possibly explain differences in values of the response variable; Ex. exam version.

Factor Level =

• a single category of the factor

• MATH: It’s the TYPES Ex. (Version 1, Version 2, Version 3)

k =

• # of factor levels

• VID: Ex. There are Version 1, Version 2, and Version 3 of the test. They are factor levels. So, k is 3.

In a CRD (see definition below), a factor level is a

Treatment.

Response Variable =

• quantitative variable being measured in each factor level. We wish to

compare the group means for this variable.

• VID: A quantitative variable whose values are of the primary interest; Ex. X = test score (75, 63, 91 on Version 1 of Exam)

Experimental Unit =

• item from which a measurement of the response variable is obtained. It is

not the units of measurement. Think of “Participants” as experimental units, for example.

• The “things” or “material” to which the experiment is applied and from which the sample data is collected– rats, company divisions, fiscal quarters, plots of tillable ground, people, etc.

Grand Mean =

• average of all values of the response variable included in the data set

• X bar bar

Grand Mean Equation

n1 * x bar 1 / n1 +…

nT =

• total # of data values

• VID: total number of observations

SSTotal =

• SSB + SSW

• Sum of Squares Between + Sum of Squares Within

SSB also called

SSTr (Sum of Squares Treatments)

SSW also called

SSE (Sum of Squares Error)

Degrees of Freedom =

# squared deviations in SS minus # averages used

Mean Squares =

Sum of squares ÷ correct df. The mean squares (MS) measurements are true

variance estimates.

F-ratio =

a ratio of two variance estimates (ratio of two MS values). This identifies how many

times bigger one estimate is than the other.

Completely Randomized Design (CRD) =

• random selection of observations that are randomly assigned to the treatments.

• VID: In a One-way ANOVA, when the factor levels (ex. Version 1, Version 2, Version 3) are assigned to the experimental units (ex. people/participants) randomly we have a completely randomized design (CRD).

• Factor level = treatment

Balanced Design =

equal number of observations in each treatment (n1 = n2 = … = nk)

Pooled Variance Estimate (see formula sheet) =

MSE

Residual =

𝑥𝑖𝑗 − 𝑥̅𝑖 = “observed – expected” = actual data value – average of group

Required Data Conditions PT1

o Independent SRSs

• Simple random samples from independent populations (groups or categories).

o Each treatment population is Normal

• The response variable (Ex. X = test scores) can be modeled well with a Normal distribution in all populations.

Required Data Conditions PT 2

o Homogeneity of variance; all treatment populations have the same value for σ2

• The response variable (Ex. X = test scores) has the same variance in all populations

• Additional is Response variable (X) is quantitative.

Reject the null hypothesis of equal means

if the F-ratio is significantly large (which will also result in the P-value being significantly small).

Reject

P-value < α

Fobs > F*

If the ANOVA test rejects the null hypothesis of equal means,

perform a post hoc analysis of the differences in means.

Tukey Method: PT 1

o All possible pairwise comparisons

o Two-sample comparisons assuming equal population variances.

Tukey Method PT 2

o Requires the Studentized Range value as a critical value (q*). The q* is corrected to control the family significance level.

If the interval contains 0,

there is no significant difference between those two means.

if C = margin of error ≥ C, there is

no significant difference between those two

means.

Family Significance Level –

• the probability of at least on Type I error among multiple simultaneous comparisons

• the significance level for a conclusion derived from multiple individual hypothesis tests.

Family significance level =

1 – (1 – α ind)^c

Family confidence level =

(1 – α ind)^c

VIDEO

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Analysis of Variance (ANOVA)

• used to determine whether multiple population means are equal or not

• done by partitioning (dividing into parts) total variation into explained and unexplained.

ni

number of observations in the ith factor level (in one factor level. can be added up to find nT)

Sum of Squares SS

• Total of squared deviations from average.

Variance /Mean of Squares (MS)

• SS divided by its degrees of freedom. The result is _.

Test Statistic, F Ratio, and F obs are

THE SAME

Homogeneity of Variance

The response variable (Ex. X = test scores) has the same variance in all

populations

F Distribution PT 1

• Bell curve is skewed to the right. (Space ON the right side)

• Line is F (v1,v2). Has significance level/a on it (Reject will go out to the right of it

F Distribution PT 2

• ν1 = numerator df = D1

• ν2 = denominator df = D2

• Underneath line is ANOTHER line, which has F*/critical value.

i is

columns/categories. Also, associated with k.

j is

rows. Also, associated with n.

F* found using:

α (in one tail)

Numerator df = dfB

Denominator df = dfW

a individual / individual significance level =

• P(Type I Error) for each separate test

• Type I Error: Reject null when null is true.

Family α / Family significance level =

• P(at least 1 Type I Error among all individual tests)

c =

# of comparisons being done

# of unique pairs (order not relevant)

c is found by =

k(k − 1)/2