Tests for the difference between two Normal meansEdexcel A-Level Further Maths: Mind map

Exam questions on Tests for the difference between two Normal means

1. Independent random samples are taken from two Normal populations. Sample $X$: $n_x=20$, $\bar x=52.8$, population variance $\sigma_x^2=9$. Sample $Y$: $n_y=30$, $\bar y=51.2$, population variance $\sigma_y^2=16$. Both variances are known.
   Test, at the $5\%$ significance level, whether the mean of $X$ is greater than the mean of $Y$. State your conclusion in context.2 marks
2. Reaction times, in milliseconds, are measured for two independent groups, $A$ and $B$. Each reaction time is Normally distributed with standard deviation $40$. Group $A$ has $36$ people with mean time $312$. Group $B$ has $49$ people with mean time $298$. A researcher tests whether the mean reaction times of the two populations differ, at the $5\%$ level.
   State the conclusion of the test, with a reason, in context.2 marks
3. Two classes take the same task. The times, in minutes, are not assumed to be Normal and the population variances are unknown. Class $1$: $n_1=60$, $\bar x_1=24.6$, $s_1^2=18.4$. Class $2$: $n_2=75$, $\bar x_2=23.1$, $s_2^2=22.5$. The teacher tests whether the population mean times differ.
   Calculate the value of the test statistic for $H_0:\mu_1=\mu_2$.3 marks