Paper 2 (Higher tier)

Which of these is a random sampling method?

Describe how to select a quota sample of 40 people (20 male, 20 female) in a town centre and give one disadvantage.

Ecologists catch and tag 30 squirrels in a wood. A week later they catch 36 squirrels, of which 6 are tagged. Use the Petersen capture–recapture method to estimate the population.

State one assumption made in using this method.

Explain how to reduce non-response in a postal survey.

The table shows (data are illustrative, written for this paper):

h (cm)	Frequency
140 < h (cm) ≤ 150	6
150 < h (cm) ≤ 155	14
155 < h (cm) ≤ 160	20
160 < h (cm) ≤ 165	16
165 < h (cm) ≤ 180	9

Calculate an estimate of the mean.

Explain why your answer to (a) is an estimate.

Calculate the frequency density for each class, ready to draw a histogram, and state which class has the greatest frequency density.

Use linear interpolation to estimate the median.

Estimate the 90th percentile.

Eleven marks in a test are (data are illustrative, written for this paper): 42, 45, 47, 48, 50, 51, 52, 53, 55, 56, 59. Find the median and the interquartile range.

An outlier is a value more than 1.5 × IQR below Q1 or above Q3. Show whether there are any outliers.

Five daily temperatures (°C) (data are illustrative, written for this paper): 10, 12, 15, 17, 21. Calculate the mean and the standard deviation. (Use σ = √(Σ(x − x̄)²/n).)

A distribution has mean 32.4, median 30.0 and standard deviation 6.0. Calculate Pearson's coefficient of skewness, 3(mean − median) ÷ standard deviation, and describe the skew.

A weighted index is calculated for a family's shopping (data are illustrative, written for this paper). Rent: index 108, weight 5; Food: index 115, weight 3; Transport: index 120, weight 2. Calculate the weighted index number.

Death rates per 1,000 in Seaview and a standard population (data are illustrative, written for this paper):

Age	Rate per 1,000	Standard population
0–29	0.8	40%
30–59	4	40%
60+	35	20%

Calculate the standardised death rate for Seaview.

Explain why a histogram rather than a bar chart is used for grouped continuous data with unequal class widths.

In a group of 60 students, 35 study French, 24 study Spanish and 11 study both (data are illustrative, written for this paper). How many study neither?

A student who studies Spanish is chosen at random. Find the probability that they also study French.

Two counters are taken at random without replacement from a bag of 5 red and 3 blue counters. Calculate the probability that both are the same colour.

15% of components are faulty. A box of 12 is chosen. The number of faulty components X follows a binomial distribution B(12, 0.15). Calculate P(X = 1).

Times to complete a puzzle (minutes) are normally distributed with mean 40 and standard deviation 5 (data are illustrative, written for this paper). Use the fact that about 68% of values lie within 1 standard deviation of the mean, and about 95% within 2, to estimate the percentage between 35 and 45.

A scatter diagram of age of car (years) against value (£) shows strong negative correlation. Explain what this means and suggest one reason why the relationship may not be linear for very old cars.

Sales of ice cream for three years show a regular pattern: high in summer, low in winter. Explain how to use moving averages and a trend line to predict sales for next summer.

For the last two Quarter 4 periods, actual sales were 96 and 104 and the trend values were 81.5 and 86.0 (data are illustrative, written for this paper). The trend line gives 90.25 for next year's Quarter 4. Estimate next year's Quarter 4 sales.

The table shows the activity chosen by 240 members of a leisure centre (data are illustrative, written for this paper):

	Swim	Gym	Classes
Adults	45	60	35
Children	55	15	30

A member is chosen at random. Find (i) P(Children); (ii) P(Gym | Children); (iii) P(Adults | Swim).

Two branches recorded daily customer waiting times (minutes). Group A: 40 people, mean 12.5. Group B: 60 people, mean 15.0. Calculate the mean of the combined group.

In a survey of 132 people, 33 chose comedy (data are illustrative, written for this paper). Calculate the angle for comedy on a pie chart.

The probability that a seed germinates is 0.85. Calculate the expected number of times in 240 trials (probability 0.85).

Raw data from a survey of students' ages and hours of sleep (data are illustrative, written for this paper): (15, 8), (16, 7.5), (14, 9), (61, 7), (15, 28), (16, 8.5). Identify two values that should be checked or removed when cleaning the data, giving reasons.

Two population pyramids are described (data are illustrative, written for this paper). Country X: wide base narrowing sharply with age. Country Y: narrow base, bulge at ages 40–60, many people over 70. Compare the populations and suggest one implication for Country Y.