Edexcel A-Level Mathematics A2 S3.3 Estimation Confidence Intervals and Tests Questions

Practise estimating means and variances, forming confidence intervals and using Normal tests for claims about populations.

Exam points

calculate unbiased estimates, standard errors and sampling distributions from summary data
use confidence intervals or z-tests to assess claims about means and differences in means
decide when Normal assumptions or large-sample results justify the method used

Covered syllabus points

S3.3.1 - Concepts of standard error,S3.3.2 - Distribution of the sample mean S3.3.3 - Concept of a confidence interval S3.3.4 - Confidence limits for a Normal S3.3.5 - Hypothesis tests for the mean of a Normal distribution S3.3.6 - Central Limit theorem for sample means S3.3.7 - Hypothesis test for the difference S3.3.8 - Use of large sample results

[Maximum number: 8]

A random sample of two observations $X_{1}$ and $X_{2}$ is taken from a population with unknown mean $\mu$ and unknown variance $\sigma^{2}$

(a)

Explain what you understand by an unbiased estimator for $\mu$

Two estimators for $\mu$ are $U_{1}$ and $U_{2}$ where

U_{1}=3 X_{1}-2 X_{2} \quad \text { and } \quad U_{2}=\frac{X_{1}+3 X_{2}}{4}

[ 1 ]

An estimator for $\mu$ is unbiased if its expected value is equal to $\mu$
(1)

(b)

Show that both $U_{1}$ and $U_{2}$ are unbiased estimators for $\mu$

The most efficient estimator among a group of unbiased estimators is the one with the smallest variance.

[ 3 ]

$\mathrm{E}\left(U_{1}\right)=3 \mathrm{E}\left(X_{1}\right)-2 \mathrm{E}\left(X_{2}\right)$ or $\mathrm{E}\left(U_{2}\right)=\frac{1}{4}\left(\mathrm{E}\left(X_{1}\right)+3 \mathrm{E}\left(X_{2}\right)\right)$

M1

$\mathrm{E}\left(U_{1}\right)=3 \mu-2 \mu=\mu$ (therefore unbiased)

A1cso

$\mathrm{E}\left(U_{2}\right)=\frac{1}{4}(\mu+3 \mu)=\mu$ (therefore unbiased)

A1cso

(c)

By finding the variance of $U_{1}$ and the variance of $U_{2}$ state, giving a reason, the most efficient estimator for $\mu$ from these two estimators.

[ 4 ]

$\operatorname{Var}(U_1)=9\operatorname{Var}(X_1)+4\operatorname{Var}(X_2)=13\sigma^2$ $\operatorname{Var}(U_2)=\frac{1}{16}\operatorname{Var}(X_1)+\frac{9}{16}\operatorname{Var}(X_2)=\frac{5}{8}\sigma^2$
As $\operatorname{Var}(U_1)>\operatorname{Var}(U_2)$ , $U_2$ is the most efficient estimator for $\mu$ .
M1 for use of $a^2\operatorname{Var}(X_1)+b^2\operatorname{Var}(X_2)$ .
A1 for $13\sigma^2$ , A1 for $5\sigma^2/8$ , A1 for $U_2$ with a correct reason.

[Maximum number: 10]

A garden centre sells bags of stones and large bags of gravel.
The weight, X kilograms, of stones in a bag can be modelled by a normal distribution with unknown mean $\mu$ and known standard deviation 0.4

The stones in each of a random sample of 36 bags from a large batch is weighed. The total weight of stones in these 36 bags is found to be 806.4 kg

(a)

Find a 98% confidence interval for the mean weight of stones in the batch.

[ 4 ]

$\left[\bar{x}=\frac{806.4}{36}=\right]22.4$ $22.4\pm2.3263\times\frac{0.4}{\sqrt{36}}$ $(22.24\ldots, 22.55\ldots)$
NB answers which are awrt (22.2,22.6) gain full marks.

(b)

Explain why the use of the Central Limit theorem is not required to answer part (a)

The manufacturer of these bags of stones claims that bags in this batch have a mean weight of 22.5 kg

[ 1 ]

(c)

Using your answer to part (a), comment on the claim made by the manufacturer.

The weight, Y kilograms, of gravel in a large bag can be modelled by a normal distribution with mean 850 kg and standard deviation 5 kg

A builder purchases 10 large bags of gravel.

[ 2 ]

(d)

Find the probability that the mean weight of gravel in the 10 large bags is less than 848 kg

[ 3 ]

$\bar{Y}\sim N\left(850,\left(\frac{5}{\sqrt{10}}\right)^2\right)$ $P(\bar{Y}<848)=P\left(Z<\frac{848-850}{\frac{5}{\sqrt{10}}}\right)=P(Z<-1.26)$ =0.1038
Calculator gives 0.10295...
ALT
N(8500,250) $P(\bar{Y}<848)=P\left(Z<\frac{8480-8500}{\sqrt{250}}\right)=P(Z<-1.26)=0.1038$

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Covered syllabus points

Question 1

Question (a)

Question (b)

Question (c)

Question 2

Question (a)

Question (b)

Question (c)

Question (d)