Pearson Edexcel IAL Mathematics Unit S3: Statistics 3 Question Bank

Practise S3 statistics through Normal combinations, sampling methods, estimation, hypothesis tests, goodness of fit and regression decisions.

Covered syllabus points

S3.1 - Combinations of random variables S3.2 - Sampling S3.3 - Estimation, confidence intervals and tests S3.4 - Goodness of fit and contingency tables S3.5 - Regression and correlation

[Maximum number: 8]

At a particular supermarket, the times taken to serve each customer in a queue at a standard checkout may be modelled by a normal distribution with mean 240 seconds and standard deviation 20 seconds.

There is a queue of 3 customers at a standard checkout.
Making a reasonable assumption about the times taken to serve these customers,

(a)

find the probability that the total time taken to serve the 3 customers will be less than 11 minutes.

[ 3 ]

Let P= time to serve a customer at a standard checkout.
$Q=P_1+P_2+P_3,\quad Q\sim N(720,1200)$ $P(Q<660)=P\left(Z<\frac{660-720}{\sqrt{1200}}\right)=P(Z<-1.732\ldots)$
M1
=0.0418
Calculator gives 0.04163...
awrt 0.041 / 0.042
A1
ALT
B1 for $Q\sim N(12,\frac13)$ .
M1 for $P(Q<11)=P\left(Z<\frac{11-12}{\sqrt{1/3}}\right)$ .

(b)

State the assumption you have made in part (a)

In the supermarket there is also an express checkout, which is reserved for customers buying 10 or fewer items. The time taken to serve a customer at this express checkout may be modelled by a normal distribution with mean 100 seconds and standard deviation 8 seconds.

On a particular day Jiang has 8 items to pay for and has to choose whether to join a queue of 3 customers waiting at a standard checkout or a queue of 7 customers waiting at the express checkout.

Using a similar assumption to that made in part (a),

[ 1 ]

(c)

find the probability that the total time taken to serve the 3 customers at the standard checkout will exceed the total time taken to serve the 7 customers at the express checkout.

[ 4 ]

Let R= time to serve a customer at an express checkout.
$S=(P_1+P_2+P_3)-(R_1+\cdots+R_7),\quad S\sim N(20,1648)$
M1 A1
$P(S>0)=P\left(Z>\frac{0-20}{\sqrt{1648}}\right)=P(Z>-0.492\ldots)$
M1
=0.6879
Calculator gives 0.6888...
awrt 0.688 / 0.689
A1

[Maximum number: 15]

Luka wants to carry out a survey of students at his school.

He obtains a list of all 280 students.

(a)

Explain how he can use this list to select a systematic sample of 40 students.
(2)

Luka is trying to make his own random number table. He generates 400 digits to put in his table. Figure 1 shows the frequency of each digit in his table.

Figure 1

A test is carried out at the 10\% level of significance to see if the digits Luka generates follow a uniform distribution.

For this test $\sum \frac{(O-E)^{2}}{E}=5.9$

[ 2 ]

Randomly select a student from 1 to 7 oe

B1

Take every $7{ }^{\text {th }}$ student

B1

(2)

(b)

Determine the conclusion of this test.

The digits generated by Luka are taken two at a time to form two-digit numbers.

Figure 2 shows the frequency of two-digit numbers in his table.

Figure 2

[ 3 ]

$v=10-1=9 \quad \chi_{9}^{2}(10 \%)=14.684$

B1 B1ft

No evidence (at 10\% level of significance) that the digits generated do not
follow a uniform distribution.

dB1

(3)

(c)

Test, at the 10% level of significance, whether the two-digit numbers generated by Luka follow a uniform distribution. You should state the hypotheses, the degrees of freedom and the critical value used for this test.

There are 70 students in Year 12 at his school.

[ 8 ]

$\mathrm{H}_{0}$ : Uniform distribution is a good fit (for the two-digit numbers generated)

$\mathrm{H}_{1}$ : Uniform distribution is not a good fit (for the two-digit numbers generated)

B1

O & E & $\frac{(O-E)^{2}}{E}$ & $\frac{O^{2}}{E}$

31 & 40 & 2.025 & 24.025

49 & 40 & 2.025 & 60.025

30 & 40 & 2.5 & 22.5

42 & 40 & 0.1 & 44.1

48 & 40 & 1.6 & 57.6

200 & 200 & 8.25 & 208.25

B1

M1 A1

$\sum \frac{(O-E)^{2}}{E}=8.25 \quad$ or $\sum \frac{O^{2}}{E}-200=8.25$

A1

$\chi_{4}^{2}(10 \%)=7.779$

B1

[8.25 > 7.779] Reject $\mathrm{H}_{0}$ / Significant

M1

There is evidence to suggest the two-digit numbers generated may not follow a
uniform distribution.

A1

(8)

(d)

State, giving a reason, the advice you would give to Luka regarding the use of his table of numbers for generating a simple random sample of 10 of the Year 12 students.
(2)

[ 2 ]

[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 $\mu$

B1

(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: 12]

A research student is investigating the number of children who are girls in families with 4 children.

The table below shows her results for 200 such families.

Table for Question Unit S3: Statistics 3 question 4 — Edexcel A-Level Mathematics A2

The research student suggests that a binomial distribution with $p=\frac{1}{2}$ could be a suitable model for the number of children who are girls in a family of 4 children.

(a)

Using her results and a 5% significance level, test the research student's claim. You should state your hypotheses, expected frequencies, test statistic and the critical value used.

The research student decides to refine the model and retains the idea of using a binomial distribution but does not specify the probability that the child is a girl.

[ 8 ]

$H_0$ : B(4,0.5) is a suitable model.
$H_1$ : B(4,0.5) is not a suitable model.
B1
Expected frequencies: 12.5, 50, 75, 50, 12.5.
M1 A1
$\sum \frac{(O-E)^2}{E}=\frac{(15-12.5)^2}{12.5}+\cdots+\frac{(10-12.5)^2}{12.5}$
or
$\sum \frac{O^2}{E}-N=\frac{15^2}{12.5}+\cdots+\frac{10^2}{12.5}-200$
A1
v=4
B1
$\chi^2_4(0.05)=9.488$ , critical region >9.488.
B1
Sufficient evidence to say that the research student's claim is not supported.
A1 ft
M1 M1

Figure for Answer to Question (a) — Edexcel A-Level Mathematics A2

(b)

Test, at the 5% significance level, whether using a binomial distribution is suitable to model the number of children who are girls in a family of 4 children.
You should state your hypotheses and the critical value used.

[ 4 ]

B1
v=3 $\chi^2_3(0.05)=7.815$ , critical region >7.815.
B1 ft
No significant evidence to say that the binomial is not a reasonable model.

All question bank results loaded

Covered syllabus points

Unit S3: Statistics 3 question 1

Question (a)

Question (b)

Question (c)

Unit S3: Statistics 3 question 2

Question (a)

Question (b)

Question (c)

Question (d)

Unit S3: Statistics 3 question 3

Question (a)

Question (b)

Question (c)

Unit S3: Statistics 3 question 4

Question (a)

Question (b)