Hypothesis tests for the difference between two meansEdexcel International A Level Further Maths: Subtopic test

1. 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.

   (a)

   Under $H_0$, what is the variance of $\bar X-\bar Y$?

   [1 mark]

   • A$1.8$
   • B$0.2$
   • C$0.911$
   • D$1.894$

   (b)

   What is the value of the test statistic?

   [1 mark]

   • A$2.3$
   • B$1.71$
   • C$1.28$
   • D$2.41$

   (c)

   Complete the test and state your conclusion in context.

   [2 marks]

   Total for question 1: 4 marks

2. 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 $B$.

   (a)

   Which pair of hypotheses should the gardener use?

   [1 mark]

   • A$H_0:\mu_A=\mu_B,\ H_1:\mu_A\neq\mu_B$
   • B$H_0:\mu_A>\mu_B,\ H_1:\mu_A=\mu_B$
   • C$H_0:\mu_A=\mu_B,\ H_1:\mu_A>\mu_B$
   • D$H_0:\mu_A-\mu_B=0.8,\ H_1:\mu_A-\mu_B>0.8$

   (b)

   What is the standard error of $\bar X_A-\bar X_B$?

   [1 mark]

   • A$0.291$
   • B$0.763$
   • C$0.381$
   • D$0.539$

   (c)

   Carry out the test and state your conclusion in context.

   [2 marks]

   Total for question 2: 4 marks

3. 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.

   (a)

   State the approximate distribution of $\bar X_P-\bar X_Q$ under $H_0$, and justify why this distribution may be used.

   [3 marks]

   (b)

   Carry out the test and state your conclusion in context.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A company trains new staff using method $A$ or method $B$, with trainees randomly assigned to a method. The times taken, in minutes, to complete a task are Normally distributed, with known standard deviations of $3.0$ for $A$ and $4.0$ for $B$. A random sample of $15$ trainees taught by $A$ has a mean time of $24.5$ minutes and a random sample of $12$ trainees taught by $B$ has a mean time of $27.8$ minutes. The company believes that method $A$ gives a shorter mean time.

   (a)

   Test, at the $5\%$ significance level, whether method $A$ gives a shorter mean time than method $B$.

   [6 marks]

   (b)

   (i) Show that the result is also significant at the $1\%$ level.
   (ii) A manager concludes: 'This proves that method $A$ is faster for every trainee.' Evaluate this conclusion, referring to the assumptions made in the test.

   [6 marks]

   Total for question 4: 12 marks

End of questions