AP/ECON 3210C Assignment Supplement: t-tests
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
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.
|
X1 |
X2 |
X3 |
Y |
Mean |
66.76 |
69.43 |
67.43 |
70.94 |
Std |
20.78 |
17.98 |
17.76 |
18.65 |
n |
99 |
99 |
99 |
99 |
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
1. Test Statistic
• F = MST / MSE
— MST is Mean Square for Treatment
— MSE is Mean Square for Error
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
2023-12-26