Tests for the difference between two Normal meansEdexcel A-Level Further Maths: Subtopic test

1. 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.

   (a)

   Find the standard error of $\bar X-\bar Y$.

   [1 mark]

   • A$0.707$
   • B$0.983$
   • C$5$
   • D$0.992$

   (b)

   Calculate the test statistic for the null hypothesis $H_0:\mu_x=\mu_y$.

   [1 mark]

   • A$1.61$
   • B$2.19$
   • C$0.32$
   • D$0.99$

   (c)

   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]

   Total for question 1: 4 marks

2. 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.

   (a)

   Which pair of hypotheses should be used?

   [1 mark]

   • A$H_0:\bar x_A=\bar x_B$, $H_1:\bar x_A\ne\bar x_B$
   • B$H_0:\mu_A=\mu_B$, $H_1:\mu_A\ne\mu_B$
   • C$H_0:\mu_A=\mu_B$, $H_1:\mu_A>\mu_B$
   • D$H_0:\mu_A-\mu_B=14$, $H_1:\mu_A-\mu_B\ne14$

   (b)

   Find the $p$-value of the test.

   [1 mark]

   • A$0.055$
   • B$0.889$
   • C$0.111$
   • D$0.945$

   (c)

   State the conclusion of the test, with a reason, in context.

   [2 marks]

   Total for question 2: 4 marks

3. 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.

   (a)

   Calculate the value of the test statistic for $H_0:\mu_1=\mu_2$.

   [3 marks]

   (b)

   Carry out the test at the $5\%$ significance level, and explain why a Normal distribution can be used here even though the population variances are unknown.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two suppliers make cables. The breaking strength, in newtons, of supplier $P$'s cables is Normally distributed with standard deviation $12$. For supplier $Q$ it is Normally distributed with standard deviation $15$. A random sample of $25$ cables from $P$ has mean strength $187.4$ and a random sample of $40$ cables from $Q$ has mean strength $181.2$.

   (a)

   Test, at the $5\%$ significance level, whether the mean breaking strength of $P$'s cables is greater than that of $Q$'s cables. State your hypotheses and conclusion.

   [6 marks]

   (b)

   (i) Find the smallest difference $\bar x_P-\bar x_Q$ that would lead to $H_0$ being rejected at the $5\%$ level.
   (ii) Find the smallest difference that would lead to rejection at the $1\%$ level, and hence comment on the strength of the evidence from the observed difference of $6.2$.

   [6 marks]

   Total for question 4: 12 marks

End of questions