Difference between two means: pooled t-testEdexcel A-Level Further Maths: Subtopic test

1. 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 $s^2$.

   (a)

   How many degrees of freedom does the $t$ statistic for testing the difference between the two population means have?

   [1 mark]

   • A$22$
   • B$20$
   • C$21$
   • D$19$

   (b)

   Find $s^2$, the pooled estimate of the common variance.

   [1 mark]

   • A$4.8$
   • B$4.47$
   • C$2.22$
   • D$4.92$

   (c)

   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]

   Total for question 1: 4 marks

2. 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.

   (a)

   Find the critical value that the test statistic should be compared with.

   [1 mark]

   • A$\pm1.771$
   • B$\pm2.179$
   • C$\pm2.160$
   • D$\pm2.131$

   (b)

   Which pair of assumptions is needed to justify this test?

   [1 mark]

   • AThe populations are Normal with the same (unknown) variance, and the samples are independent
   • BThe sample sizes are equal and the population variances are known
   • CThe population means are equal and the sample sizes are large
   • DThe sample variances are equal and the populations are Normal

   (c)

   Calculate the value of the test statistic.

   [2 marks]

   Total for question 2: 4 marks

3. 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.

   (a)

   Find the pooled estimate of the common variance of the two populations.

   [3 marks]

   (b)

   Test, at the $5\%$ significance level, whether the mean times for the two methods differ. State your hypotheses and your conclusion in context.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A gardener compares two fertilisers. Nine plots are treated with fertiliser $X$ and eleven plots with fertiliser $Y$. The yields (kg per plot) have sample means $\bar{x}=4.82$ and $\bar{y}=4.35$, and unbiased sample variances $s_x^2=0.36$ and $s_y^2=0.49$. Yields may be assumed to be independent and Normally distributed with a common variance.

   (a)

   Test, at the $5\%$ significance level, whether fertiliser $X$ gives a greater mean yield than fertiliser $Y$. Show your working and state your conclusion in context.

   [6 marks]

   (b)

   Find a $95\%$ confidence interval for $\mu_X-\mu_Y$, and state what it suggests about the difference between the fertilisers.

   [6 marks]

   Total for question 4: 12 marks

End of questions