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## One-Way ANOVA

A. One way Anova will be used a statistical test that compares the difference in the group means of a categorical dependent variable against the continuous independent variable. In this case race is the categorical independent variable with 3 levels (white, African Americans and Others), while SAT_R_pre is the continuous independent variable.
B.
 ANOVA Stanford Achievement Test NCE scores in Reading Before Sum of Squares df Mean Square F Sig. Between Groups 2341.061 2 1170.530 3.426 .036 Within Groups 42360.337 124 341.616 Total 44701.397 126

Interpretations: “A one-way between subjects ANOVA was conducted to compare the difference in the Pre-SAT reading scores among different races. There was a statistical significant difference in the Pre-SAT reading scores of the three different race with (F=3.426, p=0.036).

C. t tests - Means: Difference between two independent means (two groups)

Analysis: A priori: Compute required sample size

Input: Tail(s) = Two

Effect size d = 0.5

α err prob = 0.05

Power (1-β err prob) = 0.80

Allocation ratio N2/N1 = 1

Output: Noncentrality parameter δ = 2.8284271

Critical t = 1.9789706

Df = 126

Sample size group 1 = 64

Sample size group 2 = 64

Total sample size = 128

Actual power = 0.8014596

Using G-power analysis the required sample size is 64 for group 1 and 64 for group 2 which will amount to 128 total sample size.

## Two-Way ANOVA

A. Two- Way ANOVA is the appropriate statistical level to use. The two-way ANOVA is used to estimate how the mean of a quantitative variable changes according to the levels of two categorical variables. In this question both race and group are the categorical variables while the Pre-SAT reading scores is the dependent variable.

B.

a. R Squared = .080 (Adjusted R Squared = .042)

 Tests of Between-Subjects Effects Dependent Variable: Stanford Achievement Test NCE scores in Reading Before Source Type III Sum of Squares df Mean Square F Sig. Corrected Model 3596.055a 5 719.211 2.117 .068 Intercept 333707.428 1 333707.428 982.320 .000 group 84.123 1 84.123 .248 .620 race 2976.851 2 1488.425 4.381 .015 group * race 1064.266 2 532.133 1.566 .213 Error 41105.342 121 339.714 Total 486021.730 127 Corrected Total 44701.397 126

Interpretations: A two way analysis of variance was conducted to examine if there is a difference in Pre-SAT reading scores according to group and race. We can see from the table above that there was no statistically significant difference in mean of Pre-SAT reading scores for the two groups (F=0.248, p = .620), but there were statistically significant differences between the three race (F=4.381, p < .0005).
C. Plot of cell means
D. Post hoc test for race against Pre-SAT reading scores.
 Multiple Comparisons Dependent Variable: Stanford Achievement Test NCE scores in Reading Before Tukey HSD (I) race (J) race Mean Difference (I-J) Std. Error Sig. 95% Confidence Interval Lower Bound Upper Bound African American White -9.505* 3.6212 .026 -18.098 -.913 Others -5.740 4.9706 .482 -17.535 6.054 White African American 9.505* 3.6212 .026 .913 18.098 Others 3.765 4.7305 .706 -7.460 14.990 Others African American 5.740 4.9706 .482 -6.054 17.535 White -3.765 4.7305 .706 -14.990 7.460
Interpretations: From the table above we can see that there is a statistical significance between Pre-SAT scores of White and African Americans race with (p>0.005) while there is no significance pairs of relationship among other race.
E. F tests - ANOVA: Fixed effects, special, main effects and interactions
Analysis: A priori: Compute required sample size
Input: Effect size f = 0.8
α err prob = 0.01
Power (1-β err prob) = 0.8
Numerator df = 2
Number of groups = 2
Output: Noncentrality parameter λ = 17.2800000
Critical F = 5.5679971
Denominator df = 25
Total sample size = 27
Actual power = 0.8141890
From the G-power test above the required sample size is 27.
Correlation Analysis
A. Correlation Analysis will be used. The correlation analysis is a statistical techniques that is used to measure the relationship that exists between continuous variables.
B.
 Correlations Stanford Achievement Test NCE scores in Reading After Stanford Achievement Test NCE scores in Language After Stanford Achievement Test NCE scores in Math After Stanford Achievement Test NCE scores in Reading After Pearson Correlation 1 .737** .789** Sig. (2-tailed) .000 .000 N 148 148 148 Stanford Achievement Test NCE scores in Language After Pearson Correlation .737** 1 .738** Sig. (2-tailed) .000 .000 N 148 148 148 Stanford Achievement Test NCE scores in Math After Pearson Correlation .789** .738** 1 Sig. (2-tailed) .000 .000 N 148 148 148
**. Correlation is significant at the 0.01 level (2-tailed).
Interpretations: From the table above, we found a strong significant relationship between Stanford Achievement Test NCE scores in Reading After and Stanford Achievement Test NCE scores in Language After with (r=0.737, p<0.05). Similarly, there is a strong significant positive relationship between Stanford Achievement Test NCE scores in Reading After and Stanford Achievement Test NCE scores in Math After with (r=0.789, p>0.05). Lastly, there is strong significant relationship between Stanford Achievement Test NCE scores in language After and Stanford Achievement Test NCE scores in Math After with (r=0.738, p<0.05).
C. t tests - Correlation: Point biserial model
Analysis: A priori: Compute required sample size
Input: Tail(s) = Two
Effect size |ρ| = 0.8
α err prob = 0.05
Power (1-β err prob) = 0.8
Output: Noncentrality parameter δ = 3.5276684
Critical t = 2.5705818
Df = 5
Total sample size = 7
Actual power = 0.8029379
From the G-power test above, the total sample size is 7.
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