CAIE A-Level Mathematics A2 6.4 Sampling & Estimation

[Maximum number: 4]

The lengths, $X \mathrm{~cm}$ , of a sample of 100 insects of a certain type were summarised as follows.

n=100 \quad \sum x=36.8 \quad \sum x^{2}=17.34

(a)

Calculate unbiased estimates for the population mean and variance of X.

[ 3 ]

$\operatorname{est}(\mu)=0.368=\frac{46}{125}$
Oe.

$\operatorname{est}\left(\sigma^{2}\right)=\frac{100}{99}\left(\frac{17.34}{100}-\right.$ their $\left.^{\prime} 0.368^{\prime 2}\right)$ or $\frac{1}{99}\left(17.34-\frac{36.8^{2}}{100}\right)$
For use of a correct formula (ft their $\mu$ ).

=0.0384(3 sf)

(b)

State a necessary condition for the estimates found in part (a) to be reliable.

[ 1 ]

[Maximum number: 6]

Henri wants to choose a random sample from the 804 students at his college. He numbers the students from 1 to 804 and then uses random numbers generated by his calculator. The first 20 random digits produced by his calculator are as follows.

\begin{array}{llllllllllllllllllll} 5 & 6 & 7 & 1 & 0 & 9 & 8 & 4 & 3 & 1 & 0 & 9 & 6 & 6 & 5 & 0 & 2 & 1 & 7 & 6 \end{array}

Henri's first two student numbers are 567 and 109.

(a)

Use Henri's digits to find the numbers of the next two students in the sample.

There were 30 students in Henri's sample. He asked each of them how much time, X hours, they spent on social media each week, on average. He summarised the results as follows.

n=30 \quad \Sigma x=610 \quad \Sigma x^{2}=12405

[ 2 ]

(b)

Use this information to calculate an unbiased estimate of the mean of X and show that an unbiased estimate of the variance of X is less than 0.1 .

[ 3 ]

$\operatorname{Est}(\mu)=\frac{610}{30}$ or $\frac{61}{3}$
OE or 20.3.

$\operatorname{Est}\left(\sigma^{2}\right)=\frac{30}{29}\left(\frac{12405}{30}-\left(\frac{610}{30}\right)^{2}\right)$ or $\frac{1}{29}\left(12405-\frac{610^{2}}{30}\right)$
Use of correct formula.

=0.0575(3 sf)
Marking guidance:

Accept $\frac{5}{87}$ .

(c)

Henri's friend claims that Henri has probably made a mistake in his calculation of $\Sigma x$ or $\Sigma x^{2}$ .

Use your answer to part (b) to comment on this claim.

[ 1 ]

Variance is [unrealistically] small so Henri has [probably] made a mistake/claim is [probably] correct

B1 FT

Need both parts.

Need 'small' OE, not just < 0.1. FT their < 0.1 variance value (not-ve), e.g. 0.0556 (if omit $\frac{30}{29}$ ).

Marking guidance:

Accept 's.d. =0.24 is small, so Henri has probably made a mistake'.

Note: 'mean is large/small' scores B0, but 'mean large compared to variance so Henri prob made a mistake' scores B1.

[Maximum number: 3]

In a large population, the systolic blood pressure (SBP) of adults is normally distributed with mean 125.4 and standard deviation 18.6.

The SBP of 12-year-old children in the same population is normally distributed with mean 117. Of these children 88\% have SBP more than 108.

Find the standard deviation of this distribution.

Three adults are chosen at random from this population.

$\frac{108-117}{\sigma}=-1.175$ $1.1749<z \leqslant 1.175$ or $-1.175 \leqslant z<-1.1749$
108 and 117 substituted in ± standardisation formula, no continuity
correction, not $\sigma^{2}, \sqrt{ } \sigma$ , equated to a z-value.

$\sigma=7.66$ $7.659 \leqslant \sigma \leqslant 7.66$
If M0 scored, SC B1 for $7.659 \leqslant \sigma \leqslant 7.66$

[Maximum number: 1]

The lengths, $X \mathrm{~cm}$ , of a sample of 100 insects of a certain type were summarised as follows.

n=100 \quad \sum x=36.8 \quad \sum x^{2}=17.34

State a necessary condition for the estimates found in part (a) to be reliable.

[Maximum number: 2]

Henri wants to choose a random sample from the 804 students at his college. He numbers the students from 1 to 804 and then uses random numbers generated by his calculator. The first 20 random digits produced by his calculator are as follows.

\begin{array}{llllllllllllllllllll} 5 & 6 & 7 & 1 & 0 & 9 & 8 & 4 & 3 & 1 & 0 & 9 & 6 & 6 & 5 & 0 & 2 & 1 & 7 & 6 \end{array}

Henri's first two student numbers are 567 and 109.

Use Henri's digits to find the numbers of the next two students in the sample.

There were 30 students in Henri's sample. He asked each of them how much time, X hours, they spent on social media each week, on average. He summarised the results as follows.

n=30 \quad \Sigma x=610 \quad \Sigma x^{2}=12405

[Maximum number: 3]

A random variable X has the distribution N(410,400).
Find the probability that the mean of a random sample of 36 values of X is less than 405.

[Maximum number: 1]

The widths， $w \mathrm{~cm}$ ，of a random sample of 150 leaves of a certain kind were measured．The sample mean of w was found to be 3.12 cm ．

Using this sample，an approximate 95 % confidence interval for the population mean of the widths in centimetres was found to be［3．01，3．23］．

Explain whether it was necessary to use the Central Limit theorem in your answer to part（a）．［1］

[Maximum number: 3]

The numbers of green sweets in 200 randomly chosen packets of Frutos are summarised in the table.

Number of green sweets	0	1	2	3	>3
Number of packets	32	50	97	21	0

Calculate an unbiased estimate for the population mean of the number of green sweets in a packet of Frutos, and show that an unbiased estimate of the population variance is 0.783 correct to 3 significant figures.

$\hat{\mu}=\frac{307}{200}$ or 1.535
Marking guidance:

Accept 3sf if nothing better seen (1.53 or 1.54).

$\Sigma x^{2} f=627, \operatorname{Est}\left(\sigma^{2}\right)=\frac{200}{199}\left(\frac{\prime 627^{\prime}}{200}-{ }^{\prime} 1.535^{\prime 2}\right)$ or $\frac{1}{199}\left({ }^{\prime} 627^{\prime}-\frac{{ }^{\prime} 307^{\prime 2}}{200}\right)$
Use of a correct formula with their values.

=0.783
AG Correctly obtained with no errors seen.

[Maximum number: 3]

The widths， $w \mathrm{~cm}$ ，of a random sample of 150 leaves of a certain kind were measured．The sample mean of w was found to be 3.12 cm ．

Using this sample，an approximate 95 % confidence interval for the population mean of the widths in centimetres was found to be［3．01，3．23］．

Calculate an estimate of the population standard deviation．

$3.12+z \times \frac{\sigma}{\sqrt{150}}=3.23$
OE - correct expression. Any z, but must be a z value.

z=1.96
$\sigma=0.687$ (3sf) [cm]

[Maximum number: 4]

A student wishes to estimate the proportion, p, of students at her college who have exactly one brother. She surveys a random sample of 50 students at her college and finds that 18 of them have exactly one brother. She calculates an approximate $\alpha \%$ confidence interval for p and finds that the lower limit of the confidence interval is 0.244 correct to 3 significant figures.

Find $\alpha$ correct to the nearest integer.

$\frac{18}{50}-z \times \sqrt{\frac{\frac{18}{50} \times\left(1-\frac{18}{50}\right)}{50}}=0.244$
Use of correct equation.

z=1.709 or 1.708
Marking guidance:

Accept 1.71 if nothing better seen.

$\phi^{-1}\left({ }^{\prime} 1.709^{\prime}\right)=0.956 ; 1-2\left(1-{ }^{\prime} 0.956^{\prime}\right)$

[= 0.912]
Attempt area above or below their 1.709 and use correct method to find $\alpha$ .

$\alpha=91$
Allow $\alpha=91 \%$

0.91 or 91.2 score A0.

CAIE A-Level Mathematics A2 6.4 Sampling and Estimation Questions

Exam points

Covered syllabus points

Covered syllabus points

Question 1

Question (a)

Question (b)

Question 2

Question (a)

Question (b)

Question (c)

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Question 10