Difference between two means: pooled t-testEdexcel A-Level Further Maths: Mind map

Exam questions on Difference between two means: pooled t-test

1. Independent random samples are taken from two Normal populations whose variances are equal but unknown. Sample $X$ has size $10$ and sample variance $s_x^2=3.6$. Sample $Y$ has size $12$ and sample variance $s_y^2=6.0$. The pooled estimate of the common variance is $s^2$.
   Find the estimated standard error of $\bar{X}-\bar{Y}$, that is $s\sqrt{\frac{1}{10}+\frac{1}{12}}$, to 3 significant figures.2 marks
2. A pooled $t$ test of $H_0:\mu_X=\mu_Y$ is carried out using independent random samples of sizes $n_X=6$ and $n_Y=9$ from two Normal populations. The sample means are $\bar{x}=52.4$ and $\bar{y}=49.1$, and the pooled estimate of the common variance is $s^2=11.7$. The test is two-tailed at the $5\%$ significance level.
   Calculate the value of the test statistic.2 marks
3. Students solve the same puzzle using one of two methods. Eight students use method A, with times $x$ seconds, where $\sum x=96$ and $\sum x^2=1180$. Ten different students use method B, with times $y$ seconds, where $\sum y=150$ and $\sum y^2=2310$. Times for each method may be assumed to be Normally distributed with a common variance.
   Find the pooled estimate of the common variance of the two populations.3 marks