Hypothesis tests for the difference between two meansEdexcel International A Level Further Maths: Mind map

Exam questions on Hypothesis tests for the difference between two means

1. Two machines, $X$ and $Y$, fill bags with sugar. The masses, in grams, are Normally distributed, with known standard deviations of $4$ for machine $X$ and $5$ for machine $Y$. A random sample of $20$ bags from $X$ has a mean mass of $502.1$ and a random sample of $25$ bags from $Y$ has a mean mass of $499.8$. A test of $H_0:\mu_X=\mu_Y$ against $H_1:\mu_X\neq\mu_Y$ is carried out at the $5\%$ significance level.
   Complete the test and state your conclusion in context.2 marks
2. Seedlings are grown using either fertiliser $A$ or fertiliser $B$. The heights are Normally distributed, with known standard deviations of $2.1$ cm for $A$ and $2.4$ cm for $B$. A random sample of $30$ seedlings grown with $A$ has a mean height of $14.6$ cm and a random sample of $40$ seedlings grown with $B$ has a mean height of $13.8$ cm. A gardener wants to test, at the $5\%$ significance level, whether fertiliser $A$ gives a greater mean height than fertiliser $B$.
   Carry out the test and state your conclusion in context.2 marks
3. A supermarket chain compares customer spending at two stores, $P$ and $Q$. A random sample of $60$ customers at $P$ spent a mean of £42.50 with sample variance $81$, and a random sample of $50$ customers at $Q$ spent a mean of £39.80 with sample variance $64$. The distributions of spending are not assumed to be Normal. The chain tests $H_0:\mu_P=\mu_Q$ against $H_1:\mu_P\neq\mu_Q$ at the $5\%$ significance level.
   State the approximate distribution of $\bar X_P-\bar X_Q$ under $H_0$, and justify why this distribution may be used.3 marks