CAIE A-Level Mathematics A2 6.4 Sampling and Estimation Questions

Practise selecting random samples, calculating unbiased estimates and sampling distributions and constructing or interpreting confidence intervals for population means.

Exam points

distinguish a population parameter from a sample statistic and justify random selection
calculate unbiased estimates of population mean and variance from raw or summarised data
form a confidence interval as estimate ± critical value × standard error and interpret it

Covered syllabus points

6.4.1—Samples and populations 6.4.2—Sampling methods 6.4.3—Sample mean 6.4.4—Normal distribution 6.4.5—Sample mean 6.4.6—Unbiased estimates 6.4.7—Mean confidence intervals 6.4.8—Proportion confidence intervals

[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 $7.659 \leqslant \sigma \leqslant 7.66$

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Covered syllabus points

Question 1

Question (a)

Question (b)

Question 2

Question (a)

Question (b)

Question (c)

Question 3