Further Statistics 1: Central limit theoremEdexcel A-Level Further Maths: Topic test

1. 1
   The masses of parcels handled by a courier firm have mean $2.4$ kg and standard deviation $0.9$ kg. The distribution of the masses is not known. A random sample of $36$ parcels is taken and the sample mean mass is $\bar{X}$.

   (a)

   Which is the approximate distribution of $\bar{X}$?

   [1 mark]

   • A$\mathrm{N}(2.4,\,0.81)$
   • B$\mathrm{N}(2.4,\,0.0225)$
   • C$\mathrm{N}(2.4,\,0.15)$
   • D$\mathrm{N}(86.4,\,29.16)$

   (b)

   What is $\mathrm{P}(\bar{X}>2.55)$, to 3 significant figures?

   [1 mark]

   • A$0.159$
   • B$0.841$
   • C$0.434$
   • D$0.0228$

   (c)

   Explain why the Central Limit Theorem can be used here even though the distribution of the masses is not known.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The lifetime of a certain type of bulb has mean $800$ hours and standard deviation $120$ hours. The distribution of lifetimes is not known. A random sample of $64$ bulbs is taken and $\bar{X}$ is the sample mean lifetime, in hours.

   (a)

   What is the standard deviation of $\bar{X}$?

   [1 mark]

   • A$120$
   • B$1.875$
   • C$15$
   • D$225$

   (b)

   What is $\mathrm{P}(\bar{X}<780)$, to 3 significant figures?

   [1 mark]

   • A$0.909$
   • B$0.0478$
   • C$0.434$
   • D$0.0912$

   (c)

   Find the smallest sample size $n$ for which the standard deviation of $\bar{X}$ is less than $10$ hours.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The number of faults in a roll of fabric, $X$, has a Poisson distribution with mean $0.8$. A random sample of $60$ rolls is taken, and $\bar{X}$ is the sample mean number of faults per roll.

   (a)

   Write down the mean and the variance of $X$ and hence state the approximate distribution of $\bar{X}$.

   [3 marks]

   (b)

   Use your answer to part (a) to estimate $\mathrm{P}(\bar{X}>1)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A hiker needs $X$ attempts to light a camping stove, where $X$ is the number of attempts up to and including the first success and each attempt succeeds with probability $0.2$, independently of other attempts. A campsite has $80$ hikers, whose numbers of attempts are independent, each distributed as $X$. Let $\bar{X}$ be the mean of the $80$ values and $S$ their total.

   (a)

   (i) Find $\mathrm{E}(X)$ and $\mathrm{Var}(X)$. (ii) State the approximate distribution of $\bar{X}$. (iii) Find $\mathrm{P}(\bar{X}<4.5)$. Marks: (i) 2, (ii) 2, (iii) 2.

   [6 marks]

   (b)

   (i) Use the Central Limit Theorem to estimate $\mathrm{P}(S>420)$. (ii) Find the value of $k$, to the nearest integer, such that $\mathrm{P}(S>k)=0.01$. Marks: (i) 3, (ii) 3.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A random variable $X$ has the binomial distribution $\mathrm{B}(8,0.25)$. It is observed independently $45$ times, and $\bar{X}$ is the mean of the $45$ observations.

   (a)

   What is the approximate variance of $\bar{X}$?

   [1 mark]

   • A$1.5$
   • B$\frac{1}{30}$
   • C$\frac{2}{45}$
   • D$0.224$

   (b)

   What is $\mathrm{P}(\bar{X}>2.2)$, to 3 significant figures?

   [1 mark]

   • A$0.137$
   • B$0.863$
   • C$0.435$
   • D$0.171$

   (c)

   The number of observations is increased from $45$ to $180$. Describe, with justification, the effect on the standard deviation of $\bar{X}$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The masses of adult passengers using a lift have mean $74$ kg and standard deviation $11$ kg. The lift carries $36$ adult passengers, whose masses may be modelled as a random sample from this population. Let $T$ be the total mass of the $36$ passengers, in kg.

   (a)

   Which is the approximate distribution of $T$?

   [1 mark]

   • A$\mathrm{N}(74,\,121)$
   • B$\mathrm{N}(2664,\,121)$
   • C$\mathrm{N}(2664,\,66)$
   • D$\mathrm{N}(2664,\,4356)$

   (b)

   What is the probability that the total mass exceeds $2800$ kg, to 3 significant figures?

   [1 mark]

   • A$0.980$
   • B$0.0394$
   • C$0.0197$
   • D$0.366$

   (c)

   State two assumptions that are needed for the calculation in part (b) to be valid.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   In a table tennis club, the number of games $X$ that a player must play to win $3$ games has a negative binomial distribution with $r=3$ and $p=0.4$, where each game is won with probability $0.4$ independently of other games. A random sample of $50$ players is taken, and $\bar{X}$ is the mean of their values of $X$.

   (a)

   Find $\mathrm{E}(X)$ and $\mathrm{Var}(X)$.

   [3 marks]

   (b)

   Use the Central Limit Theorem to estimate $\mathrm{P}(\bar{X}>8)$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The lengths of pieces of wire cut by a machine have mean $25$ cm and standard deviation $4$ cm, with an unknown distribution. A batch of $100$ pieces is selected at random and the batch mean length is $\bar{X}$ cm.

   (a)

   (i) State the approximate distribution of $\bar{X}$ and justify it. (ii) Find $\mathrm{P}(24.2<\bar{X}<25.8)$ to 4 decimal places. (iii) Find the value $a$ such that $\mathrm{P}(|\bar{X}-25|<a)=0.9$. Marks: (i) 2, (ii) 2, (iii) 2.

   [6 marks]

   (b)

   A batch is accepted when its mean length lies between $24.2$ cm and $25.8$ cm. Five independent batches of $100$ pieces are inspected. Using the probability found in part (a)(ii), find the probability that (i) all five batches are accepted, (ii) exactly four of the five batches are accepted. Give your answers to 3 significant figures. Marks: (i) 3, (ii) 3.

   [6 marks]

   Total for question 8: 12 marks

End of questions