Further Statistics 2: Confidence intervals and tests using the t distributionEdexcel A-Level Further Maths: Topic test

1. 1
   A random sample of $12$ batteries of one brand has lifetimes, in hours, that may be modelled by a Normal distribution. The sample mean is $48.2$ and the unbiased estimate of the population standard deviation is $s=3.6$.

   (a)

   How many degrees of freedom does the $t$ distribution used for a confidence interval for the population mean have?

   [1 mark]

   • A$12$
   • B$10$
   • C$11$
   • D$22$

   (b)

   What critical value of $t$ is used for a $95\%$ two-sided confidence interval for the population mean?

   [1 mark]

   • A$2.201$
   • B$1.796$
   • C$2.179$
   • D$1.960$

   (c)

   Find a $95\%$ confidence interval for the population mean lifetime.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Two machines fill bags of rice. The bag masses from each machine are Normally distributed with the same unknown variance. A random sample of $8$ bags from machine $P$ has an unbiased variance estimate of $4.2$ g$^2$, and an independent random sample of $10$ bags from machine $Q$ has an unbiased variance estimate of $3.0$ g$^2$.

   (a)

   How many degrees of freedom does the $t$ distribution for a pooled test of the difference between the two means have?

   [1 mark]

   • A$18$
   • B$17$
   • C$7$
   • D$16$

   (b)

   What is the pooled estimate $s_p^2$ of the common variance?

   [1 mark]

   • A$3.6$
   • B$3.525$
   • C$3.533$
   • D$7.2$

   (c)

   State two assumptions that must be made about the populations for a pooled $t$ test to be valid.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A physiotherapist measures the range of movement of the shoulder, in degrees, of $9$ patients before and after a course of treatment. Let $d$ be the increase in range (after minus before) for each patient. The differences may be assumed to be a random sample from a Normal distribution, and the summary data are $\sum d=54$ and $\sum d^2=400$.

   (a)

   Find the mean $\bar d$ and an unbiased estimate of the variance of the differences.

   [3 marks]

   (b)

   Test, at the $5\%$ significance level, whether the treatment increases the mean range of movement.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A bottler fills jars with honey on two production lines. The jars are labelled $500$ g. The masses on each line are Normally distributed and the two lines have the same unknown variance. A random sample of $9$ jars from line $A$ has $\sum x=4527$ and $\sum x^2=2277249$. An independent random sample of $11$ jars from line $B$ has $\sum y=5456$ and $\sum y^2=2706406$, all masses in grams.

   (a)

   Test, at the $5\%$ significance level, whether the mean mass of jars from line $A$ differs from $500$ g.

   [6 marks]

   (b)

   Test, at the $5\%$ significance level, whether the mean masses of jars from the two lines differ.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A café owner suspects that a coffee machine, which is set to dispense $200$ ml per cup, actually dispenses too little. The volumes dispensed are Normally distributed. The owner measures $10$ cups, finding a sample mean of $197.4$ ml and an unbiased estimate of the standard deviation of $3.9$ ml. A one-tailed test is carried out at the $5\%$ significance level.

   (a)

   Which pair of hypotheses is appropriate for the owner's test?

   [1 mark]

   • A$H_0:\mu=200,\ H_1:\mu\neq200$
   • B$H_0:\mu=200,\ H_1:\mu<200$
   • C$H_0:\mu=200,\ H_1:\mu>200$
   • D$H_0:\mu=200,\ H_1:\bar x<200$

   (b)

   What is the critical region for the test statistic $t$?

   [1 mark]

   • A$t>1.833$
   • B$t<-2.262$
   • C$t<-1.812$
   • D$t<-1.833$

   (c)

   Given that the critical region is $t<-1.833$, carry out the test and state the conclusion in context.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Two brands of paint are compared. Drying times, in minutes, are Normally distributed with equal unknown variances. A random sample of $7$ tins of brand $X$ has mean drying time $42.0$ and an independent random sample of $9$ tins of brand $Y$ has mean drying time $38.5$. The pooled estimate of the common variance is $s_p^2=6.25$. A two-tailed test of $H_0:\mu_X=\mu_Y$ is carried out at the $5\%$ significance level.

   (a)

   What is the standard error of $\bar X-\bar Y$, using the pooled variance estimate?

   [1 mark]

   • A$1.26$
   • B$1.59$
   • C$0.50$
   • D$2.50$

   (b)

   What is the value of the test statistic?

   [1 mark]

   • A$1.40$
   • B$2.20$
   • C$2.78$
   • D$0.36$

   (c)

   Given that the critical values for this test are $\pm2.145$, state the conclusion of the test in context.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A dietitian compares total cholesterol levels, in mmol dm$^{-3}$, after two diets. The levels are Normally distributed with the same unknown variance for both diets. Eight volunteers follow diet $P$, giving a mean of $5.9$ and an unbiased variance estimate of $0.36$. An independent group of ten volunteers follows diet $Q$, giving a mean of $5.4$ and an unbiased variance estimate of $0.30$.

   (a)

   Find the pooled estimate of the common variance and state the number of degrees of freedom used in a pooled test.

   [3 marks]

   (b)

   Hence find a $95\%$ confidence interval for $\mu_P-\mu_Q$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A trainer records the 100 m times, in seconds, of $6$ sprinters before and after a training programme. Before: $11.8,\ 12.1,\ 11.6,\ 12.4,\ 11.9,\ 12.2$. After: $11.6,\ 11.9,\ 11.5,\ 12.1,\ 11.8,\ 11.9$. The times may be assumed to be Normally distributed, and the differences (before minus after) are also Normally distributed.

   (a)

   Carry out a paired $t$ test at the $1\%$ significance level to determine whether the training programme reduces the mean time.

   [6 marks]

   (b)

   A colleague analyses the same times as two independent samples, using a pooled $t$ test of the same hypotheses at the $1\%$ level, with sample variances $0.084$ for before and $0.048$ for after. Carry out this test and explain why the paired test is the more appropriate analysis.

   [6 marks]

   Total for question 8: 12 marks

End of questions