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AP/ECON 3210C Assignment Supplement: t-tests

Part 1 of your assignment is based on your background, not directly studied in this course. I assume most of you know it but still we may need to refresh our memory.  Here are examples oft-tests.

In the Part 1 of your assignment:

#2 is descriptive statistics.  You have to discuss some natures and distributions of the data, including mean, sd, and boxplot or other forms of graphs.

#3 is one sample t test.

#4 is a two-sample t test.

#5 is a test of equality of multiple means based on ANOVA.

One sample t test

In your assignment, you have two samples (data of two different years), and lets say X1 is the first year data and Y is the second year data. And we carry on the one sample tteste for X1. Then the null hypothesis, Ho, is µ = 70.94 (this is the mean of the second year data)

has the tdistribution with degrees of freedom df = n –  1.

Ho: µ = 70.94 vs Ha: µ≠70.94 thenp-value = 0.048→ Reject Ho when α = 0.05

Ho: µ = 70.94 vs Ha: µ> 0.94 thenp-value = 0.9759→ Do not reject Ho when α = 0.05

Ho: µ = 70.94 vs Ha: µ< 0.94 thenp-value = 0.024→ Reject Ho when α = 0.05

Two-sample t test

If σ1  ≠ σ2  assumed

Ho: μ1  =  μ2  vs Ha: μ1   ≠  μ2  thenp-value = 0.14→ Do not reject Ho when α = 0.05

Ho: μ1  =  μ2  vs Ha: μ1   >  μ2  thenp-value = 0.93→ Do not reject Ho when α = 0.05

Ho: μ1  =  μ2  vs Ha: μ1   < μ2 thenp-value = 0.067→ Do not reject Ho when α = 0.05

If σ1  = σ2  assumed

ANOVA analysis:

Treatment Variation:

Random (Error) Variation

ANOVA F-Test Test Statistic

2.   Degrees of Freedom

• n1  = k –  1 numerator degrees of freedom

• n2  = n – k denominator degrees of freedom

— k = Number of groups

— n = Total sample size

ANOVA Summary Table

ANOVA F-Test Critical Value: Fα,k-1,n-k, Always One-Tail

ANOVA F-Test to Compare k Treatment Means: Completely Randomized Design

H0 : µ1  = µ2  =  … = µk

Ha : At least two treatment means differ

Test Statistic: F = MST/MSE

Rejection region: F > Fa, where Fa is based on  (k –  1) numerator degrees of freedom (associated with MST) and  (n – k) denominator degrees of freedom (associated with MSE) .

Conditions Required for a Valid ANOVA F-test: Completely Randomized Design

1.   The samples are randomly selected in an independent manner from the k treatment

populations.  (This can be accomplished by randomly assigning the experimental units to the treatments.)

2.   All k sampled populations have distributions that are approximately normal.

3.   The k population variances are equal  (i.e.,

ANOVA F-Test Hypotheses