Pearson Edexcel IAL Mathematics Unit S1: Statistics 1 Question Bank

Practise S1 statistics using data summaries, probability models, regression, random variables and normal distribution calculations.

Covered syllabus points

S1.1 - Mathematical models in probability and statistics S1.2 - Representation and summary of data S1.3 - Probability S1.4 - Correlation and regression S1.5 - Discrete random variables S1.6 - The Normal distribution

[Maximum number: 14]

Statistical models are a cheap and quick way to make predictions about real-world situations.

Madison wants to develop a model to describe the relationship between the average daily temperature, $t^\circ \mathrm C$ , and a household's daily gas consumption, $d\ \mathrm{m^3}$ , in winter.

Madison takes a random sample of 12 days in winter and codes the daily gas consumption so that

w=11.5d

These data are summarised as follows

S_{tt}=26.43\quad S_{tw}=-91.55\quad \sum w=339.25\quad \sum t=9.1\quad \sum w^2=10036.45

(a)

Give one other reason why statistical models are used.

[ 1 ]

(b)

Show that $\mathrm{S}_{w w}=445.57$ to 2 decimal places.

[ 1 ]

$S_{ww}=10036.45-\frac{339.25^2}{12}=445.57$

B1*

(c)

Find the value of $\mathrm{S}_{t d}$ and the value of $\mathrm{S}_{d d}$

[ 3 ]

$S_{td}=\frac{-91.55}{11.5}$

or

$S_{td}=14.4-\frac{29.5\times9.1}{12}$

or

$S_{dd}=\frac{445.57}{11.5^2}$

or

$S_{dd}=75.9-\frac{29.5^2}{12}$

M1

$S_{td}=-7.9608\ldots$

A1

$S_{dd}=3.3691\ldots$

A1

(d)

Find the product moment correlation coefficient between d and t

[ 2 ]

$r=\frac{-91.55}{\sqrt{445.57\times26.43}}$

or

$r=\frac{-7.9608\ldots}{\sqrt{3.3691\ldots\times26.43}}$

M1

$r=-0.8436\ldots$ $r=-0.844\ \text{(awrt)}$

A1

(e)

Give an interpretation, in context, of your product moment correlation coefficient.

[ 1 ]

(f)

Show that the equation of the regression line of w on t is

w=-3.46 t+30.9

where the values of the intercept and the gradient are given to 3 significant figures.

[ 3 ]

$b=\frac{-91.55}{26.43}=-3.4638\ldots$

M1

$a=\frac{339.25}{12}-(-3.4638\ldots)\frac{9.1}{12}=30.897\ldots$

M1

w=-3.46t+30.9

A1*

(g)

Write down an equation of the regression line of d on t

[ 1 ]

(h)

Using your equation in part (g)

[ 2 ]

(i)

estimate the daily gas consumption in winter when the temperature is $2^{\circ} \mathrm{C}$

[ 1 ]

$2.0852 \ldots\left(\mathrm{~m}^{3}\right)$

B1

(ii)

interpret the effect an increase of $1^{\circ} \mathrm{C}$ in average daily temperature is expected to have on the daily gas consumption in winter.

[ 1 ]

The gas consumption will decrease by $0.3\ \mathrm{m^3}$ .

B1ft

[Maximum number: 13]

The histogram shows the times taken, in seconds, by each of 260 people to complete a puzzle.

Figure for Question Unit S1: Statistics 1 question 2 — Edexcel A-Level Mathematics AS

(a)

Use the histogram to complete the frequency table for the times taken to complete the puzzle.

Table for Question (a) — Edexcel A-Level Mathematics AS

Given that $\sum \mathrm{f} t=11087.5$ and $\sum \mathrm{f} t^{2}=505718.75$

[ 3 ]

(b)

find an estimate for

[ 3 ]

(i)

the mean time taken to complete the puzzle,

[ 1 ]

(ii)

the standard deviation of the times taken to complete the puzzle.

[ 2 ]

(c)

Use linear interpolation to estimate the median time taken to complete the puzzle.

[ 2 ]

(d)

Describe the skewness of these data. Give a reason for your answer.

Three of the 260 people are chosen at random.

[ 1 ]

(e)

Estimate the probability that all 3 of their times are less than 36 seconds.

[ 4 ]

[Maximum number: 11]

The Venn diagram shows the events A, B, C and D, where p, q, r and s are probabilities.

Figure for Question Unit S1: Statistics 1 question 3 — Edexcel A-Level Mathematics AS

(a)

Write down the value of

[ 3 ]

(i)

P(A)

[ 1 ]

(ii)

$\mathrm{P}(A \mid B)$

[ 1 ]

$P(A\mid B)=1.$

B1

(iii)

$P(A\mid C)$

[ 1 ]

$P(A\mid C)=0.$

B1

(b)

Given that $P(B'\cap D')=\frac{7}{10}$ and $P(C\mid D)=\frac{3}{5}$ , find the exact value of q and the exact value of r.

[ 6 ]

$\frac{q}{q+r}=\frac35$ $0.13+p+s=\frac{7}{10}$ p+q+r+s+0.12+0.13=1
Solving gives
$q=0.108,\quad r=0.072.$

M1 M1 M1 dM1 A1 A1

(c)

Given also that $P(B\cup C')=\frac{5}{8}$ , find the exact value of s.

[ 2 ]

$\frac58=0.13+0.12+0.072+s$ s=0.303.

M1 A1

[Maximum number: 10]

Students on a psychology course were given a pre-test at the start of the course and a final exam at the end of the course. The teacher recorded the number of marks achieved on the pre-test, p, and the number of marks achieved on the final exam, f, for 34 students and displayed them on the scatter diagram.

The equation of the least squares regression line for these data is found to be
f=10.8+0.748p

For these students, the mean number of marks on the pre-test is 62.4

Considering the equation of the regression line, Priya says that she would expect someone who scored 0 marks on the pre-test to score 10.8 marks on the final exam.

Later the teacher discovers an error in the recorded data. The student who achieved a score of 98 on the pre-test, scored 92 not 29 on the final exam.
The summary statistics used for the model f=10.8+0.748p are corrected to include this information and a new least squares regression line is found.
Given the original summary statistics were,
n=34
sum p=2120
sum pf=133486
Spp=15573.76
Spf=11648.35

(a)

Use the regression model to find the mean number of marks on the final exam.

[ 2 ]

(b)

Give an interpretation of the gradient of the regression line.

[ 1 ]

(a)

For each additional mark scored on the pre-test, the average mark on the final exam
increases by 0.748

Figure for Answer to Question (b) — Edexcel A-Level Mathematics AS

B1 (1)

M1
$\mathrm{Al}^{\mathrm{P}_{3}} \mathrm{O}_{2}$
(2)
B1
(1)

(c)

Comment on the reliability of Priya's statement.

[ 1 ]

(d)

Write down the number of marks achieved on the final exam for the student who exceeded the expectation of the regression model by the largest number of marks.

[ 1 ]

(e)

calculate the gradient of the new regression line. Show your working clearly.

[ 5 ]

$S_{pp}=15573.76$ (no change)
$\sum pf=133486-2842+9016=139660$ $\sum pf$ increases by 98(92-29)=6174 $\sum f=57.47\times34+(92-29)\approx2017$ $\frac{\sum p\sum f}{n}$ increases by $\frac{2120(92-29)}{34}=3928.235\ldots$ $S_{pf}$ increases by $6174-3928.235=2245.764\ldots$ $b=\frac{11648.35+2245.764\ldots}{15573.76}=0.89\ldots$
awrt 0.9

M1 M1 dM1 M1 A1

All question bank results loaded

Covered syllabus points

Unit S1: Statistics 1 question 1

Question (a)

Question (b)

Question (c)

Question (d)

Question (e)

Question (f)

Question (g)

Question (h)

Question (i)

Question (ii)

Unit S1: Statistics 1 question 2

Question (a)

Question (b)

Question (i)

Question (ii)

Question (c)

Question (d)

Question (e)

Unit S1: Statistics 1 question 3

Question (a)

Question (i)

Question (ii)

Question (iii)

Question (b)

Question (c)

Unit S1: Statistics 1 question 4

Question (a)

Question (b)

Question (c)

Question (d)

Question (e)