Question

Suppose that two different sets of treatments are of interest. The first
treatment has two levels ( A versus B ). The second treatment has two levels ( a
versus b ). Denote by y the response variable of interest.
a) Construct a statistical test for testing whether there is a difference in expected
value of y between A and B.
b) Construct a statistical test for testing whether there is a difference in expected
value of y among the four treatment combinations Aa, Ab, Ba, Bb.

Answer 1

(a)

Construct a statistical test for testing whether there is a difference in expected
value of y between A and B.

Two Independent Samples t test

H0: Null Hypothesis: ( There is no significant difference in expected value of y between A and B. )

HA: Alternative Hypothesis: ( There is a significant difference in expected value of y between A and B. ) (Claim)

Procedure:
From the given data, obtain the Test Statistic as follows:

where s_P is the pooled standard deviation.

From the given significace level and degrees of freedom n_A + n_B - 2, get the calculate of critical values of t.

If the calculated value of t is less than critical value of t, the difference is not significant. Fail to reject null hypothesis.

Conclusion:
The data do not support the claim that there is a significant difference in expected value of y between A and B.

If the calculated value of t is greater than critical value of t, the difference is significant. Reject null hypothesis.

Conclusion:
The data do support the claim that there is a significant difference in expected value of y between A and B.

(b)

Construct a statistical test for testing whether there is a difference in expected value of y among the four treatment combinations Aa, Ab, Ba, Bb.

One way ANOVA test

H0: Null Hypothesis: (There is no difference in expected value of y among the four treatment combinations Aa, Ab, Ba, Bb. )

HA: Alternative Hypothesis: (There is a difference in expected value of y among the four treatment combinations Aa, Ab, Ba, Bb. ) (Claim)

Procedure:
From the given data, complete the ANOVA Table:

Source of Variation	Sum of Squares	Degrees of Freedom	Mean Square	F
Treatments	SST	k-1
Error	SSE	N - k
Total	SS	N - 1

From the given value of significance level = and degrees of freedom for numerator and denominaor, get the value of critical value of F.

If the calculated value of F is less than critical value of F, the difference is not significant. Fail to reject null hypothesis.

Conclusion:
The data do not support theclaim that there is a difference in expected value of y among the four treatment combinations Aa, Ab, Ba, Bb.

If the calculated value of F is greater than critical value of F, the difference is significant. Reject null hypothesis.

Conclusion:
The data support theclaim that there is a difference in expected value of y among the four treatment combinations Aa, Ab, Ba, Bb.