Skewness

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.

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.

Over a long period of time a small company recorded the amount it received in sales per month. The results are summarised below.

An outlier is an observation that falls either 1.5 × interquartile range above the upper quartile or 1.5 × interquartile range below the lower quartile.

On the graph paper below, draw a box plot to represent these data, indicating clearly any outliers.

State the skewness of the distribution of the amount of sales received. Justify your answer.

The company claims that for 75 % of the months, the amount received per month is greater than £10 000. Comment on this claim, giving a reason for your answer.

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 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.