Measures of location

All the students in the year-group took Test 1, but some students missed Test 2.

State the number of students who missed Test 2.

Find an estimate of the number of students whose marks for Test 1 were in the range 45 to 65

inclusive.

The final parts of the graphs are horizontal.

For Test 1, explain what this means about the results.

It is required that the number of students obtaining grade A* should be the same on both tests. The minimum mark for grade A* on Test 2 was 76.

Find the minimum mark for grade A* on Test 1.

A teacher commented that the median marks on both tests were very similar. This suggests that, on average, the levels of difficulty on both tests were about the same.

Use the cumulative frequency diagram to make another comparison between Test 1 and Test

2 about the easier questions on both papers.

The marks for Test 1 are summarised in the table.

Find estimates of the mean and standard deviation of the Test 1 marks.

Use your answers to part (f)(i) to comment on whether there may be any outliers

amongst the Test 1 marks.

The stem-and-leaf diagram shows the heights, in centimetres, of 15 plants.

| 0 | 2 | 1 | 0 | 2 | 4 | 3 | 0 2 4 9 | 4 | 1 2 4 7 9 | 5 | 3 7 | 6 | 2

Key: | 2 | 5 means 25 cm.

Draw a box-and-whisker plot to illustrate the data.

A statistician intends to analyse the data, but wants to ignore any outliers before doing so.

Discuss briefly whether there are any heights in the diagram which the statistician should

ignore.

The heights, in centimetres, of a random sample of 150 plants of a certain variety were measured. The results are summarised in the histogram.

Frequency

One of the 150 plants is chosen at random, and its height, X cm, is noted.

Show that P(20 < X < 30) = 0.147, correct to 3 significant figures.

Sam suggests that the distribution of X can be well modelled by the distribution N(40, 100).

Give a brief justification for the use of the normal distribution in this context.

Give a brief justification for the choice of the parameter values 40 and 100.

Use Sam’s model to find P (20 < X < 30).

Nina suggests a different model. She uses the midpoints of the classes to calculate estimates, m and s, for the mean and standard deviation respectively, in centimetres, of the 150 heights. She then uses the distribution N(m, s²) as her model.

Use Nina’s model to find P (20 < X < 30).

Complete the table in the Printed Answer Booklet to show the probabilities obtained

from Sam’s model and Nina’s model.

By considering the different ranges of values of X given in the table, discuss how well

the two models fit the original distribution.

The table shows the age structure of usual residents of 18 Local Authorities (LAs) in the North West region of the UK in 2011.

Without reference to any other columns, explain how you would use only the columns for the age ranges 0 to 17 and 18 to 24 to decide whether an LA might be one of the following.

An LA that includes a university

An LA that attracts young couples to live

An LA that attracts retired people to live

Using your answers to part (a), identify the following.

Four LAs that might include a university

Three LAs that might be attractive to retired people

Explain why your answer to part (b)(ii), based only on the columns for the age ranges 0 to 17

and 18 to 24, may not be reliable.

The lower quartile, median and upper quartile of the percentages in the column “Age 65 and over” are 14.56%, 15.99% and 16.76% respectively.

Use this information to comment on your answers to part (b)(ii) and part (c).

In a magazine article, a councillor plans to describe a typical LA in the North West region. He wants to quote the average percentage of residents aged 65 or over.

The mean of the percentages in the column “Age 65 and over” is 16.90%.

Use this information, and the information given in part (d), to explain whether the median or

the mean better represents the data in the column “Age 65 and over”.

As part of a research project, the masses, m grams, of a random sample of 1000 pebbles from a certain beach were recorded. The results are summarised in the table.

Calculate estimates of the mean and standard deviation of these masses.

The masses, x grams, of a random sample of 1000 pebbles on a different beach were also found. It was proposed that the distribution of these masses should be modelled by the random variable X ∼ N(200, 3600).

Use the model to find P (150 < X < 210).

Use the model to determine x₁ such that P(160 < X < x₁) = 0.6, giving your answer correct

to five significant figures.

It was found that the smallest and largest masses of the pebbles in this second sample were 112 g and 288 g respectively.

Use these results to show that the model may not be appropriate.

Suggest a different value of a parameter of the model in the light of these results.