Statistics 3 (S3): Estimation, confidence intervals and testsEdexcel International A Level Further Maths: Topic test

1. 1
   A random sample of 5 observations of a variable $X$ gives the values $4$, $7$, $9$, $12$ and $13$.

   (a)

   Find an unbiased estimate of the population variance $\sigma^2$.

   [1 mark]

   • A$10.8$
   • B$54$
   • C$13.5$
   • D$3.67$

   (b)

   Which statement is true of the estimator $\frac{1}{n}\sum\left(X_i-\bar X\right)^2$ for $\sigma^2$?

   [1 mark]

   • AIt is biased, because its expected value is less than $\sigma^2$
   • BIt is unbiased, because every observation is used
   • CIt is biased, because its expected value is greater than $\sigma^2$
   • DIt is unbiased only when $n$ is small

   (c)

   The observations come from a population with standard deviation $4$. Find the standard error of the sample mean for a sample of size $5$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The heights of a species of tree are Normally distributed with mean $12$ m and standard deviation $3$ m. Random samples of $9$ trees are taken.

   (a)

   What is the distribution of the sample mean $\bar X$?

   [1 mark]

   • A$\mathrm{N}(12,\,9)$
   • B$\mathrm{N}(12,\,1)$
   • C$\mathrm{N}(12,\,3)$
   • D$\mathrm{N}\left(12,\,\frac13\right)$

   (b)

   Find $\mathrm{P}(\bar X>13.5)$.

   [1 mark]

   • A$0.3085$
   • B$0.1587$
   • C$0.9332$
   • D$0.0668$

   (c)

   Find the probability that a single tree chosen at random is shorter than $10$ m.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The lifetime, in hours, of a type of rechargeable battery is Normally distributed with standard deviation $25$. A random sample of $16$ batteries has a mean lifetime of $480$ hours.

   (a)

   Find a $95\%$ confidence interval for the mean lifetime of this type of battery.

   [3 marks]

   (b)

   The manufacturer claims that the mean lifetime is $500$ hours. State, with a reason, whether the interval in part (a) supports the claim. Another sample with the same mean is taken, and the manufacturer wants a $95\%$ confidence interval of width less than $10$ hours. Find the smallest sample size needed.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A student takes $10$ independent random samples, each of size $25$, from a Normal population with known standard deviation $12$. From each sample the student constructs a $95\%$ confidence interval for the population mean $\mu$.

   (a)

   Calculate the width of each interval. Explain what is meant by saying that each interval is a $95\%$ confidence interval, and state the expected number of the $10$ intervals that contain $\mu$. Find the probability that all $10$ intervals contain $\mu$.

   [6 marks]

   (b)

   For one of the samples, the sample mean is $143.6$. Find a $95\%$ confidence interval for $\mu$ from this sample. Use your interval to comment on the claim that $\mu=150$, stating the link with a hypothesis test. If $\mu$ were $150$, state the distribution of the sample mean for samples of size $25$ and find the probability that the sample mean is at most $143.6$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A bakery claims that the mean mass of its loaves is $800$ g. The masses are Normally distributed with standard deviation $12$ g. An inspector weighs a random sample of $36$ loaves, finds a sample mean of $796$ g, and tests at the $5\%$ significance level whether the mean mass is less than $800$ g.

   (a)

   Find the value of the test statistic.

   [1 mark]

   • A$-0.33$
   • B$-2.00$
   • C$-0.67$
   • D$-1.00$

   (b)

   Which is the correct conclusion of the test?

   [1 mark]

   • ADo not reject $\mathrm{H}_0$, because $-2.00$ is greater than $-1.645$
   • BDo not reject $\mathrm{H}_0$, because a difference of $4$ g is too small to matter
   • CReject $\mathrm{H}_0$; it is proved that the mean mass is $796$ g
   • DReject $\mathrm{H}_0$; there is evidence at the $5\%$ level that the mean mass is less than $800$ g

   (c)

   Find the critical region for the sample mean $\bar X$ in this test.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The annual income of households in a district has a positively skewed distribution with unknown mean $\mu$. A researcher takes a random sample of $120$ households and finds a sample mean of £$34\,500$ and a sample standard deviation of £$9\,600$. She tests $\mathrm{H}_0:\mu=33\,000$ against $\mathrm{H}_1:\mu>33\,000$ at the $5\%$ significance level.

   (a)

   Why can the sample mean $\bar X$ be treated as approximately Normally distributed here?

   [1 mark]

   • ABy the Central Limit theorem, because the sample is large
   • BBecause the incomes in the population are Normally distributed
   • CBecause the sample standard deviation equals the population standard deviation
   • DBecause the sample mean is an unbiased estimate of $\mu$

   (b)

   Which value should the test statistic be compared with?

   [1 mark]

   • A$1.282$
   • B$1.960$
   • C$1.645$
   • D$2.326$

   (c)

   Find the value of the test statistic.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A physiotherapist compares the recovery times, in days, of patients after two treatments, $P$ and $Q$. A random sample of $50$ patients given $P$ had a mean recovery time of $21.4$ days with sample standard deviation $5.2$ days. An independent random sample of $40$ patients given $Q$ had a mean of $19.6$ days with sample standard deviation $4.8$ days.

   (a)

   Find the value of the test statistic for a test of whether the mean recovery times for the two treatments are different.

   [3 marks]

   (b)

   Using your answer to part (a), carry out the test at the $5\%$ significance level, stating your hypotheses and your conclusion in context.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The breaking strengths, in newtons, of cables made by two firms $A$ and $B$ are Normally distributed with standard deviations $14$ and $10$ respectively. A random sample of $12$ cables from $A$ has mean breaking strength $215.4$, and an independent random sample of $15$ cables from $B$ has mean $208.9$.

   (a)

   Test, at the $5\%$ significance level, whether the mean breaking strength of cables from $A$ is greater than that of cables from $B$.

   [6 marks]

   (b)

   Find a $95\%$ confidence interval for $\mu_A-\mu_B$ and say what it shows about a two-tailed test of $\mu_A=\mu_B$ at the $5\%$ level. Explain why the strengths needing to be Normally distributed matters for this method.

   [6 marks]

   Total for question 8: 12 marks

End of questions