Question Type 14: Calculating the critical value and/or p-value to conclude the two sample t-test Exercises

Samples: Sample 1 ($n_1=9$, $\bar x_1=50$, $s_1=6$), Sample 2 ($n_2=9$, $\bar x_2=45$, $s_2=7$). At 5% (two-tailed), find the critical $t$-value and conclusion for testing $H_0:\mu_1=\mu_2$.

Two small samples: Sample 1 ($n_1=15$, $\bar x_1=7.2$, $s_1=1.5$), Sample 2 ($n_2=15$, $\bar x_2=6.8$, $s_2=1.2$). At the 5% level, test $H_0:\mu_1=\mu_2$ vs $H_1:\mu_1>\mu_2$. Find the critical $t$ and conclude.

A researcher compares the average reaction times of two groups. Sample 1: $n_1=12$, $\bar x_1=24$ ms, $s_1=4$ ms; Sample 2: $n_2=15$, $\bar x_2=20$ ms, $s_2=5$ ms. At the 5% significance level, test for a difference in means by finding the critical $t$-value and stating your conclusion.

Two independent samples yield: Sample 1: $n_1=20$, $\bar x_1=15$, $s_1=3$; Sample 2: $n_2=18$, $\bar x_2=12$, $s_2=4$. Test $H_0:\mu_1=\mu_2$ vs $H_1:\mu_1>\mu_2$ at the 5% level. Calculate the $t$-statistic, the one-tailed p-value, and state your conclusion.

Two larger samples: Sample 1 ($n_1=30$, $\bar x_1=54$, $s_1=8$), Sample 2 ($n_2=28$, $\bar x_2=50$, $s_2=7$). At the 5% level (two-tailed), find the critical $t$-value and conclude if the means differ.

Two small equal-size samples: Sample 1 ($n_1=12$, $\bar x_1=45$, $s_1=5$), Sample 2 ($n_2=12$, $\bar x_2=40$, $s_2=4$). Compute the two-tailed p-value for $H_0:\mu_1=\mu_2$ at 5% and conclude.

Samples of exam scores: Sample 1 ($n_1=25$, $\bar x_1=100$, $s_1=12$), Sample 2 ($n_2=30$, $\bar x_2=95$, $s_2=10$). At the 10% level, test if the mean of group 1 differs from group 2. Compute the two-tailed p-value and conclude.

Industrial lifetimes: Sample 1 ($n_1=18$, $\bar x_1=120$, $s_1=15$), Sample 2 ($n_2=16$, $\bar x_2=110$, $s_2=20$). Test $H_0:\mu_1=\mu_2$ vs $H_1:\mu_1>\mu_2$ at 5%. Compute the one-tailed p-value and conclusion.

Two samples of weights give Sample 1: $n_1=8$, $\bar x_1=85$, $s_1=6$; Sample 2: $n_2=10$, $\bar x_2=80$, $s_2=5$. At the 1% significance level, test for a difference in means (two-tailed). Find the critical $t$-value and conclude.

Samples: Sample 1 ($n_1=14$, $\bar x_1=5.4$, $s_1=0.8$), Sample 2 ($n_2=16$, $\bar x_2=5.1$, $s_2=0.9$). Test $H_0:\mu_1=\mu_2$ vs $H_1:\mu_1<\mu_2$ at 1%. Calculate the p-value and conclude.

Sample 1 ($n_1=16$, $\bar x_1=250$, $s_1=30$), Sample 2 ($n_2=14$, $\bar x_2=230$, $s_2=25$). Test $H_0:\mu_1=\mu_2$ vs $H_1:\mu_1\neq\mu_2$. Compute the two-tailed p-value and conclude at 5%.

Sample 1 ($n_1=20$, $\bar x_1=200$, $s_1=30$), Sample 2 ($n_2=22$, $\bar x_2=185$, $s_2=25$). At 1% level, test $H_0:\mu_1=\mu_2$ vs $H_1:\mu_1>\mu_2$. Find the critical $t$ and state your decision.