Measures of location

Runners in an athletics club can train with either coach A or coach B for the 400 m race. Coach A trains 120 runners for the 400 m and records the best time, x seconds, for each runner. The results are summarised by the following statistics

Calculate the mean of the best times for the runners trained by coach A

Calculate the standard deviation of the best times for the runners trained by coach A The mean and standard deviation for the best times of the 100 runners trained for the 400 m by coach B are 55.1 seconds and 3.6 seconds respectively. A 400 m race consists of equal numbers of the fastest runners trained by coach A and by coach B

State, giving a reason, which coach is more likely to have trained the winner.

Ming is studying the large data set for Perth in 2015 He intended to use all the data available to find summary statistics for the Daily Mean Air Temperature, x^oC. Unfortunately, Ming selected an incorrect variable on the spreadsheet. This incorrect variable gave a mean of 5.3 and a standard deviation of 12.4

Using your knowledge of the large data set, suggest which variable Ming selected. The correct values for the Daily Mean Air Temperature are summarised as n = 184 x 2801 2. x² 44 695 4.

Calculate the mean and standard deviation for these data. One of the months from the large data set for Perth in 2015 has

for Daily Mean Air Temperature.

Suggest, giving a reason, a month these data may have come from.

Ben is studying the Daily Total Rainfall, xmm, in Leeming for 1987 He used all the data from the large data set and summarised the information in the following table.

Explain how the data will need to be cleaned before Ben can start to calculate statistics such as the mean and standard deviation.

Calculate estimates for

the mean Daily Total Rainfall,

the standard deviation of the Daily Total Rainfall. Ben suggests using the statistic calculated in part (b)(i) to estimate the annual mean Daily Total Rainfall in Leeming for 1987

Using your knowledge of the large data set,

give a reason why these data would not be suitable,

state, giving a reason, how you would expect the estimate in part (b)(i) to differ from the actual annual mean Daily Total Rainfall in Leeming for 1987

Dian uses the large data set to investigate the Daily Total Rainfall, rmm, for Camborne.

Write down how a value of 0 r 0.05 is recorded in the large data set. Dian uses the data for the 31 days of August 2015 for Camborne and calculates the following statistics

Use these statistics to calculate

the mean of the Daily Total Rainfall in Camborne for August 2015,

the standard deviation of the Daily Total Rainfall in Camborne for August 2015. Dian believes that the mean Daily Total Rainfall in August is less in the South of the UK than in the North of the UK. The mean Daily Total Rainfall in Leuchars for August 2015 is 1.72 mm to 2 decimal places.

State, giving a reason, whether this provides evidence to support Dian's belief. Dian uses the large data set to estimate the proportion of days with no rain in Camborne for 1987 to be 0.27 to 2 decimal places.

Explain why the distribution B(14, 0.27) might not be a reasonable model for the number of days without rain for a 14‑day summer event.

Stav is studying the large data set for September 2015 He codes the variable Daily Mean Pressure, x, using the formula y = x − 1010 The data for all 30 days from Hurn are summarised by

State the units of the variable x

Find the mean Daily Mean Pressure for these 30 days.

Find the standard deviation of Daily Mean Pressure for these 30 days. Stav knows that, in the UK, winds circulate

• in a clockwise direction around a region of high pressure

• in an anticlockwise direction around a region of low pressure

The table gives the Daily Mean Pressure for 3 locations from the large data set on 26/09/2015

The Cardinal Wind Directions for these 3 locations on 26/09/2015 were, in random order, W NE E You may assume that these 3 locations were under a single region of pressure.

Using your knowledge of the large data set, place each of these Cardinal Wind Directions in the correct location in the table. Give a reason for your answer.

Each member of a group of 27 people was timed when completing a puzzle. The time taken, x minutes, for each member of the group was recorded. These times are summarised in the following box and whisker plot.

Find the range of the times.

Find the interquartile range of the times. For these 27 people Σ x = 607.5 and Σ x² = 17 623.25

calculate the mean time taken to complete the puzzle,

calculate the standard deviation of the times taken to complete the puzzle. Taruni defines an outlier as a value more than 3 standard deviations above the mean.

State how many outliers Taruni would say there are in these data, giving a reason for your answer. Adam and Beth also completed the puzzle in a minutes and b minutes respectively, where a b. When their times are included with the data of the other 27 people ● the median time increases ● the mean time does not change

Suggest a possible value for a and a possible value for b, explaining how your values satisfy the above conditions.

Without carrying out any further calculations, explain why the standard deviation of all 29 times will be lower than your answer to part (d).

The length of time, T, minutes, spent completing a particular task has probability density function

Use algebraic integration to find E(T)

find Var(T)

Find the cumulative distribution function F(t)

Find the 20th percentile of the time taken to complete the task.

Find the probability that the time spent completing the task is more than 1.5 minutes. Given that a person has already spent 1.5 minutes on the task,

find the probability that this person takes more than 3 minutes to complete the task.

The continuous random variable X has the following cumulative distribution function

where k, a and b are constants. Given that the mode of X is

show that a = 4

Find P(X 2.5) giving your answer to 3 significant figures.

The continuous random variable X has a probability density function

where k is a positive constant.

Sketch the graph of f (x).

Show that the value of k is

Define fully the cumulative distribution function F(x).

Hence find the 90th percentile of the distribution.

Find P[E(X) X 5.5]

A random variable X has probability density function given by

where k is a constant.

Write down the mode of X.

Specify fully the cumulative distribution function F(x).

Find the upper quartile of X.

The random variable X has probability density function f(x) given by

where k is a constant.

Sketch f (x).

Write down the mode of X.

describe, giving a reason, the skewness of the distribution.

Use integration to find the value of k.

Write down the lower quartile of X.

find the exact value of P(X 3).

In an experiment some children were asked to estimate the position of the centre of a circle. The random variable D represents the distance, in centimetres, between the child’s estimate and the actual position of the centre of the circle. The cumulative distribution function of D is given by

Find the median of D.

Find the mode of D. Justify your answer.

The experiment is conducted on 80 children.

Find the expected number of children whose estimate is less than 1 cm from the actual centre of the circle.

The random variable X has probability density function f(x) given by

where k is a constant.

Find the mode of X.

Use algebraic integration to find E(X).

By comparing your answers to parts (b) and (c),

describe the skewness of X, giving a reason for your answer.

The continuous random variable X has a cumulative distribution function

where a and b are constants.

Find the value of a and the value of b.

Use integration to find E(X).

Show that the lower quartile of X lies between 1.425 and 1.435

The continuous random variable X has the following probability density function

where a and b are constants.

find a second equation in a and b,

hence find the value of a and the value of b.

Find, to 3 significant figures, the median of X.

Comment on the skewness. Give a reason for your answer.

Figure 1 shows a sketch of the probability density function f (x) of the random variable X. For 0 x 3, f (x) is represented by a curve OB with equation f (x) = kx², where k is a constant. For 3 x a, where a is a constant, f (x) is represented by a straight line passing through B and the point (a, 0). For all other values of x, f (x) = 0. Given that the mode of X = the median of X, find

the mode,

the value of k,

the value of a.

Without calculating E(X) and with reference to the skewness of the distribution

state, giving your reason, whether E(X) 3, E(X) 3 or E(X) 3.

The continuous random variable X has probability density function given by

Sketch f (x) showing clearly the points where it meets the x-axis.

Write down the value of the mean, of X.

The cumulative distribution function of X is given by

where a is a constant.

Find the value of a.

Show that the lower quartile of X, q₁, lies between 2.29 and 2.31

Hence find the upper quartile of X, giving your answer to 1 decimal place.

Find, to 2 decimal places, the value of k so that

A continuous random variable X has the probability density function f(x) shown in Figure 1.

Show that f(x) = 4 – 8x for 0 x 0.5 and specify f(x) for all real values of x.

Find the cumulative distribution function F(x).

Find the median of X.

Write down the mode of X.

State, with a reason, the skewness of X.

The lifetime, X, in tens of hours, of a battery has a cumulative distribution function F(x) given by

Find the median of X, giving your answer to 3 significant figures.

Find, in full, the probability density function of the random variable X.

Find P(X ≥ 1.2)

A camping lantern runs on 4 batteries, all of which must be working. Four new batteries are put into the lantern.

Find the probability that the lantern will still be working after 12 hours.

The random variable Y has probability density function f(y) given by

where k and a are positive constants.

Given that E(Y) = 1.75

show that a = 4 and write down the value of k.

For these values of a and k,

sketch the probability density function,

write down the mode of Y.

The continuous random variable X has probability density function f(x) given by

where k is a constant.

Find the cumulative distribution function F(x).

Find the mean of X.

Show that the median of X lies between x = 2.6 and x = 2.7

The three independent random variables A, B and C each has a continuous uniform distribution over the interval [0, 5].

Find P(A > 3).

Find the probability that A, B and C are all greater than 3.

The random variable Y represents the maximum value of A, B and C. The cumulative distribution function of Y is

Find the probability density function of Y.

Sketch the probability density function of Y.

Write down the mode of Y.

Find E(Y).

Find P(Y > 3).

Figure 1 shows a sketch of the probability density function f(x) of the random variable X. The part of the sketch from x = 0 to x = 4 consists of an isosceles triangle with maximum at (2, 0.5).

Write down E(X).

The probability density function f(x) can be written in the following form.

Find the values of the constants a and b.

Show that , the standard deviation of X, is 0.816 to 3 decimal places.

Find the lower quartile of X.

State, giving a reason, whether P(2 – X is more or less than 0.5

The length of a telephone call made to a company is denoted by the continuous random variable T. It is modelled by the probability density function

Show that the value of k

Find P(T > 6).

Calculate an exact value for E(T) and for Var(T).

Write down the mode of the distribution of T.

It is suggested that the probability density function, f(t), is not a good model for T.

Sketch the graph of a more suitable probability density function for T.

A random variable X has probability density function given by

Show that the cumulative distribution function F(x) can be written in the form ax² + bx + c, for 1 x 4 where a, b and c are constants.

Define fully the cumulative distribution function F(x).

Show that the upper quartile of X is 2.5 and find the lower quartile.

Given that the median of X is 1.88

describe the skewness of the distribution. Give a reason for your answer.

The following grouped frequency distribution summarises the number of minutes, to the nearest minute, that a random sample of 100 motorists were delayed by roadworks on a stretch of motorway one Monday.

A histogram has been drawn to represent these data. The bar representing a delay of (3–6) minutes has a width of 2 cm and a height of 9.5 cm.

Calculate the width and the height of the bar representing a delay of (11–15) minutes.

Use linear interpolation to estimate the median delay.

Calculate an estimate of the mean delay.

Calculate an estimate of the standard deviation of the delays.

Evaluate this coefficient for the above data, giving your answer to 2 significant figures. On the following Friday, the coefficient of skewness for the delays on this stretch of motorway was – 0.22

State, giving a reason, how the delays on this stretch of motorway on Friday are different from the delays on Monday.

An estate agent is studying the cost of office space in London. He takes a random sample of 90 offices and calculates the cost, £ x per square foot. His results are given in the table

A histogram is drawn for these data and the bar representing 50 x< 60 is 2cm wide and 8cm high.

Calculate the width and height of the bar representing 20 x< 40

Use linear interpolation to estimate the median cost.

Estimate the mean cost of office space for these data.

Estimate the standard deviation for these data.

Describe, giving a reason, the skewness.

Rika suggests that the cost of office space in London can be modelled by a normal distribution with mean £50 and standard deviation £10

With reference to your answer to part (e), comment on Rika’s suggestion.

Use Rika’s model to estimate the 80th percentile of the cost of office space in London.

A midwife records the weights, in kg, of a sample of 50 babies born at a hospital. Her results are given in the table below.

Weight (w kg)	Frequency (f)	Weight midpoint (x)
0 w 2	1	1
2 w 3	8	2.5
3 w 3.5	17	3.25
3.5 w 4	17	3.75
4 w 5	7	4.5

[You may use ™Ix² = 611.375] A histogram has been drawn to represent these data. The bar representing the weight 2 ≤ w 3 has a width of 1 cm and a height of 4 cm.

Calculate the width and height of the bar representing a weight of 3 ≤ w 3.5

Use linear interpolation to estimate the median weight of these babies.

Show that an estimate of the mean weight of these babies is 3.43 kg.

Find an estimate of the standard deviation of the weights of these babies.

Shyam decides to model the weights of babies born at the hospital, by the random variable W, where W ~ N(3.43, 0.65²)

With reference to your answers to (b), (c)(i) and (d) comment on Shyam’s decision.

A newborn baby weighing 3.43 kg is born at the hospital.

Without carrying out any further calculations, state, giving a reason, what effect the addition of this newborn baby to the sample would have on your estimate of the

mean,

standard deviation.

The marks of a group of female students in a statistics test are summarised in Figure 1

Write down the mark which is exceeded by 75% of the female students.

The marks of a group of male students in the same statistics test are summarised by the stem and leaf diagram below.

Find the median and interquartile range of the marks of the male students.

An outlier is a mark that is either more than 1.5 × interquartile range above the upper quartile or more than 1.5 × interquartile range below the lower quartile.

In the space provided on Figure 1 draw a box plot to represent the marks of the male students, indicating clearly any outliers.

Compare and contrast the marks of the male and the female students.

An agriculturalist is studying the yields, y kg, from tomato plants. The data from a random sample of 70 tomato plants are summarised below.

Yield ( y kg)	Frequency (f)	Yield midpoint (x kg)
0 y < 5	16	2.5
5 y < 10	24	7.5
10 y < 15	14	12.5
15 y < 25	12	20
25 y < 35	4	30

A histogram has been drawn to represent these data. The bar representing the yield 5 y < 10 has a width of 1.5 cm and a height of 8 cm.

Calculate the width and the height of the bar representing the yield 15 y < 25

Use linear interpolation to estimate the median yield of the tomato plants.

Estimate the mean and the standard deviation of the yields of the tomato plants.

Describe, giving a reason, the skewness of the data.

Estimate the number of tomato plants in the sample that have a yield of more than 1 standard deviation above the mean.

The following table summarises the times, t minutes to the nearest minute, recorded for a group of students to complete an exam.

Estimate the mean and standard deviation of these data.

Use linear interpolation to estimate the value of the median.

Show that the estimated value of the lower quartile is 18.6 to 3 significant figures.

Estimate the interquartile range of this distribution.

Give a reason why the mean and standard deviation are not the most appropriate summary statistics to use with these data.

The person timing the exam made an error and each student actually took 5 minutes less than the times recorded above. The table below summarises the actual times.

Without further calculations, explain the effect this would have on each of the estimates found in parts (a), (b), (c) and (d).

A survey of 100 households gave the following results for weekly income £ y.

Income y (£)	Mid-point	Frequency f
0 y < 200	100	12
200 y < 240	220	28
240 y < 320	280	22
320 y < 400	360	18
400 y < 600	500	12
600 y < 800	700	8

A histogram was drawn and the class 200 y < 240 was represented by a rectangle of width 2 cm and height 7 cm.

Calculate the width and the height of the rectangle representing the class 320 y < 400

Use linear interpolation to estimate the median weekly income to the nearest pound.

Estimate the mean and the standard deviation of the weekly income for these data.

Use this measure to calculate the skewness for these data and describe its value.

Katie suggests using the random variable X which has a normal distribution with mean 320 and standard deviation 150 to model the weekly income for these data.

Find P(240 < X < 400).

With reference to your calculations in parts (d) and (e) and the data in the table, comment on Katie’s suggestion.

A policeman records the speed of the traffic on a busy road with a 30 mph speed limit. He records the speeds of a sample of 450 cars. The histogram in Figure 2 represents the results.

Calculate the number of cars that were exceeding the speed limit by at least 5 mph in the sample.

Estimate the value of the mean speed of the cars in the sample.

Estimate, to 1 decimal place, the value of the median speed of the cars in the sample.

Comment on the shape of the distribution. Give a reason for your answer.

State, with a reason, whether the estimate of the mean or the median is a better representation of the average speed of the traffic on the road.

A class of students had a sudoku competition. The time taken for each student to complete the sudoku was recorded to the nearest minute and the results are summarised in the table

(You may use fx² = 8603.75)

Write down the mid-point for the 9 - 12 interval.

Use linear interpolation to estimate the median time taken by the students.

Estimate the mean and standard deviation of the times taken by the students.

The teacher suggested that a normal distribution could be used to model the times taken by the students to complete the sudoku.

Give a reason to support the use of a normal distribution in this case.

On another occasion the teacher calculated the quartiles for the times taken by the students to complete a different sudoku and found

Describe, giving a reason, the skewness of the times on this occasion.

Keith records the amount of rainfall, in mm, at his school, each day for a week. The results are given below. 2.8 5.6 2.3 9.4 0.0 0.5 1.8 Jenny then records the amount of rainfall, x mm, at the school each day for the following 21 days. The results for the 21 days are summarised below.

Calculate the mean amount of rainfall during the whole 28 days.

Keith realises that he has transposed two of his figures. The number 9.4 should have been 4.9 and the number 0.5 should have been 5.0 Keith corrects these figures.

State, giving your reason, the effect this will have on the mean.

On a randomly chosen day, each of the 32 students in a class recorded the time, t minutes to the nearest minute, they spent on their homework. The data for the class is summarised in the following table.

Use interpolation to estimate the value of the median.

Given that

find the mean and the standard deviation of the times spent by the students on their homework.

Comment on the skewness of the distribution of the times spent by the students on their homework. Give a reason for your answer.

A teacher selects a random sample of 56 students and records, to the nearest hour, the time spent watching television in a particular week.

Hours	110	11 20	21 25	26 30	31 40	41 59
Frequency	6	15	11	13	8	3
Mid-point	5.5	15.5		28		50

Find the mid-points of the 21 25 hour and 31 40 hour groups.

A histogram was drawn to represent these data. The 11 20 group was represented by a bar of width 4 cm and height 6 cm.

Find the width and height of the 26 30 group.

Estimate the mean and standard deviation of the time spent watching television by these students.

Use linear interpolation to estimate the median length of time spent watching television by these students.

The teacher estimated the lower quartile and the upper quartile of the time spent watching television to be 15.8 and 29.3 respectively.

State, giving a reason, the skewness of these data.

A researcher measured the foot lengths of a random sample of 120 ten-year-old children. The lengths are summarised in the table below.

Foot length, l, (cm)	Number of children
10 l < 12	5
12 l < 17	53
17 l < 19	29
19 l < 21	15
21 l < 23	11
23 l < 25	7

Use interpolation to estimate the median of this distribution.

Calculate estimates for the mean and the standard deviation of these data.

One measure of skewness is given by

Evaluate this coefficient and comment on the skewness of these data.

Greg suggests that a normal distribution is a suitable model for the foot lengths of ten-year-old children.

Using the value found in part (c), comment on Greg’s suggestion, giving a reason for your answer.

In a study of how students use their mobile telephones, the phone usage of a random sample of 11 students was examined for a particular week. The total length of calls, y minutes, for the 11 students were 17, 23, 35, 36, 51, 53, 54, 55, 60, 77, 110

Find the median and quartiles for these data.

A value that is greater than Q₃ + 1.5 × (Q₃ – Q₁) or smaller than Q₁ – 1.5 × (Q₃ – Q₁) is defined as an outlier.

Show that 110 is the only outlier.

Using the graph paper on page 15 draw a box plot for these data indicating clearly the position of the outlier.

The value of 110 is omitted.

Show that S_yy for the remaining 10 students is 2966.9

These 10 students were each asked how many text messages, x, they sent in the same week. The values of S_xx and S_xy for these 10 students are S_xx = 3463.6 and S_xy = –18.3.

Calculate the product moment correlation coefficient between the number of text messages sent and the total length of calls for these 10 students.

A parent believes that a student who sends a large number of text messages will spend fewer minutes on calls.

Comment on this belief in the light of your calculation in part (e).

In a shopping survey a random sample of 104 teenagers were asked how many hours, to the nearest hour, they spent shopping in the last month. The results are summarised in the table below.

A histogram was drawn and the group (8 – 10) hours was represented by a rectangle that was 1.5 cm wide and 3 cm high.

Calculate the width and height of the rectangle representing the group (16 – 25) hours.

Use linear interpolation to estimate the median and interquartile range.

Estimate the mean and standard deviation of the number of hours spent shopping.

State, giving a reason, the skewness of these data.

State, giving a reason, which average and measure of dispersion you would recommend to use to summarise these data.

The age in years of the residents of two hotels are shown in the back to back stem and leaf diagram below. Abbey Hotel 8 5 0 means 58 years in Abbey hotel and 50 years in Balmoral hotel Balmoral Hotel

For the Balmoral Hotel,

write down the mode of the age of the residents,

find the values of the lower quartile, the median and the upper quartile.

Find the mean, , of the age of the residents.

One measure of skewness is found using

Evaluate this measure for the Balmoral Hotel.

For the Abbey Hotel, the mode is 39, the mean is 33.2, the standard deviation is 12.7 and the measure of skewness is – 0.454

Compare the two age distributions of the residents of each hotel.

Cotinine is a chemical that is made by the body from nicotine which is found in cigarette smoke. A doctor tested the blood of 12 patients, who claimed to smoke a packet of cigarettes a day, for cotinine. The results, in appropriate units, are shown below.

Find the mean and standard deviation of the level of cotinine in a patient’s blood.

Find the median, upper and lower quartiles of these data.

A doctor suspects that some of his patients have been smoking more than a packet of cigarettes per day. He decides to use Q₃+1.5(Q₃–Q₁) to determine if any of the cotinine results are far enough away from the upper quartile to be outliers.

Identify which patient(s) may have been smoking more than a packet of cigarettes a day. Show your working clearly.

Research suggests that cotinine levels in the blood form a skewed distribution.

Evaluate this measure and describe the skewness of these data.