Further Statistics 2: Estimation, confidence intervals and tests using a Normal distributionEdexcel A-Level Further Maths: Topic test

1. 1
   $X_1$ and $X_2$ form a random sample of size $2$ from a population with unknown mean $\mu$ and variance $\sigma^2$. The statistic $T=\frac{2X_1+X_2}{4}$ is proposed as an estimator of $\mu$.

   (a)

   What is $\mathrm{E}(T)$?

   [1 mark]

   • A$\mu$
   • B$3\mu$
   • C$\frac{3\mu}{4}$
   • D$\frac{\mu}{4}$

   (b)

   What is the bias of $T$ as an estimator of $\mu$?

   [1 mark]

   • A$-\frac{\mu}{4}$
   • B$\frac{\mu}{4}$
   • C$\frac{3\mu}{4}$
   • D$0$

   (c)

   Find the constant $k$ such that $kT$ is an unbiased estimator of $\mu$, and show that it is unbiased.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A botanist measures the lengths, in cm, of the stems of a random sample of $8$ plants: $21,\ 24,\ 19,\ 25,\ 22,\ 23,\ 20,\ 26$. The sample has $\sum x=180$ and $\sum x^2=4092$.

   (a)

   What is the unbiased estimate of the population variance?

   [1 mark]

   • A$5.25$
   • B$6$
   • C$42$
   • D$2.45$

   (b)

   What is the estimated standard error of the sample mean?

   [1 mark]

   • A$0.75$
   • B$2.45$
   • C$0.810$
   • D$0.866$

   (c)

   Explain why the divisor $n-1$ is used in the unbiased estimate of the population variance.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The resting heart rate, in beats per minute, of adult cyclists is Normally distributed with standard deviation $7$. A random sample of $30$ cyclists has a mean resting heart rate of $52.4$ beats per minute.

   (a)

   Find a $95\%$ confidence interval for the population mean resting heart rate.

   [3 marks]

   (b)

   A larger sample is to be taken so that a $95\%$ confidence interval has width less than $4$ beats per minute. Find the smallest sample size needed.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A paint company states that the volume of paint in its tins is Normally distributed with mean $5.00$ litres and standard deviation $0.15$ litres. An inspector takes a random sample of $25$ tins and finds a mean volume of $4.93$ litres. Assume that the standard deviation is $0.15$ litres.

   (a)

   Find a $99\%$ confidence interval for the population mean volume. Use your interval to comment on the company's claim and interpret the interval.

   [6 marks]

   (b)

   The inspector suspects that the tins contain less than $5.00$ litres on average. Test this suspicion at the $5\%$ significance level, and explain how your conclusion relates to your interval in part (a).

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Plants are grown under two lighting regimes. The heights, in cm, under LED lighting, $X$, and under fluorescent lighting, $Y$, are Normally distributed with known variances $\sigma_x^2=16$ and $\sigma_y^2=25$. Independent random samples of sizes $10$ and $20$ give sample means $\bar x=48.2$ and $\bar y=46.9$. A test is carried out of $\mathrm{H}_0:\mu_x=\mu_y$ against $\mathrm{H}_1:\mu_x>\mu_y$.

   (a)

   What is the variance of $\bar X-\bar Y$?

   [1 mark]

   • A$0.35$
   • B$41$
   • C$1.37$
   • D$2.85$

   (b)

   What is the value of the test statistic?

   [1 mark]

   • A$1.30$
   • B$0.770$
   • C$0.456$
   • D$2.20$

   (c)

   Complete the test at the $5\%$ significance level, stating your conclusion in context.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Two couriers' delivery times, in minutes, are compared using independent random samples. Courier $X$: $n_x=60$, $\bar x=34.2$, $s_x^2=36$. Courier $Y$: $n_y=80$, $\bar y=32.8$, $s_y^2=49$. The population variances are unknown. A test is carried out of $\mathrm{H}_0:\mu_x=\mu_y$ against $\mathrm{H}_1:\mu_x\neq\mu_y$.

   (a)

   Why can the Normal distribution be used for the test statistic even though the population variances are unknown?

   [1 mark]

   • AThe samples are large, so the Central Limit Theorem applies and the sample variances can replace the population variances
   • BThe population variances must be equal
   • CThe population distributions must be exactly Normal
   • DThe sample means are equal

   (b)

   What is the value of the test statistic?

   [1 mark]

   • A$1.15$
   • B$0.152$
   • C$1.27$
   • D$1.81$

   (c)

   Carry out the test at the $10\%$ significance level, stating your conclusion in context.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A laboratory weighs a reference mass four times. The readings $X_1,X_2,X_3,X_4$ in mg are independent, each distributed $\mathrm{N}(\mu,4)$ with $\mu$ unknown. Two estimators of $\mu$ are $T_1=\frac{X_1+X_2+X_3+X_4}{4}$ and $T_2=\frac{X_1+3X_2}{4}$.

   (a)

   Show that $T_2$ is an unbiased estimator of $\mu$ and find $\mathrm{Var}(T_2)$.

   [3 marks]

   (b)

   $T_1$ is also unbiased and $T_1\sim\mathrm{N}(\mu,1)$. State, with a reason, which estimator is better. Find the probability that $T_1$ is within $1.5$ mg of $\mu$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Two bakeries, $P$ and $Q$, make loaves whose masses, in grams, are Normally distributed. The standard deviation of the mass of a loaf is $9$ g for bakery $P$ and $11$ g for bakery $Q$. A random sample of $15$ loaves from $P$ has mean mass $752.4$ g and a random sample of $20$ loaves from $Q$ has mean mass $747.1$ g. The samples are independent.

   (a)

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

   [6 marks]

   (b)

   Find a $95\%$ confidence interval for $\mu_P-\mu_Q$. Explain how it agrees with your conclusion in part (a) and what it shows about the two bakeries.

   [6 marks]

   Total for question 8: 12 marks

End of questions