Question 7 of 8: Two-Sample F-test and t-test — Brick Compressive Strength

Question 7: Two-Sample F-test and t-test — Brick Compressive Strength (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Mixture A: $n_A=11$, $\bar x_A=12{,}000$, $s_A=800$. Mixture B: $n_B=12$, $\bar x_B=12{,}900$, $s_B=1{,}000$ (independent random samples, both populations assumed Normal).

Find. (a) test $H_0:\sigma_A^2=\sigma_B^2$ at $\alpha=0.05$. (b) test $H_0:\mu_A=\mu_B$ at $\alpha=0.05$, using the outcome of (a) to choose pooled vs. separate-variance $t$.

Approach. Run the F-test on the variance ratio first (it decides whether pooling is valid), then run the two-sample $t$-test on the means using the pooled-variance form since (a) does not reject equal variances.

1. (a) F-test for equal variances. Placing the larger sample variance on top, $$F=\dfrac{s_B^2}{s_A^2}=\dfrac{1{,}000^2}{800^2}=1.5625$$ with $df_1=n_B-1=11$, $df_2=n_A-1=10$. Critical values $F_{0.025,11,10}=3.665$ and $F_{0.975,11,10}=0.284$. Since $0.284\lt 1.5625\lt 3.665$, fail to reject $H_0$. $$\boxed{\text{Fail to reject } H_0:\ \text{the two variances are not significantly different}}$$
2. (b) Pooled two-sample t-test for equal means. Since (a) supports $\sigma_A^2=\sigma_B^2$, pool the variances: $$s_p^2=\dfrac{(n_A-1)s_A^2+(n_B-1)s_B^2}{n_A+n_B-2}=\dfrac{10(800)^2+11(1{,}000)^2}{21}=828{,}571.4,\quad s_p=910.3$$ $$t=\dfrac{\bar x_A-\bar x_B}{s_p\sqrt{1/n_A+1/n_B}}=\dfrac{12{,}000-12{,}900}{910.3\sqrt{1/11+1/12}}=-2.369$$ with $df=n_A+n_B-2=21$, $t_{0.025,21}=2.080$. Since $|-2.369|\gt2.080$, reject $H_0$. $$\boxed{\text{Reject } H_0:\ \text{the mean compressive strengths ARE significantly different (Mixture B is stronger)}}$$