Histograms

A medical researcher is studying the number of hours, T, a patient stays in hospital following a particular operation. The histogram on the page opposite summarises the results for a random sample of 90 patients.

Use the histogram to estimate P(10 < T < 30) For these 90 patients the time spent in hospital following the operation had

• a mean of 14.9 hours

• a standard deviation of 9.3 hours

Tomas suggests that T can be modelled by N(14.9, 9.3²)

With reference to the histogram, state, giving a reason, whether or not Tomas’ model could be suitable. Xiang suggests that the frequency polygon based on this histogram could be modelled by a curve with equation

where

• x is measured in tens of hours

• k is a constant

Use algebraic integration to show that

Show that, for Xiang’s model, k = 99 to the nearest integer.

Estimate P(10 < T < 30) using

Tomas’ model of T ~ N(14.9, 9.3²)

Xiang’s curve with equation y = 99xe^–x and the answer to part (c) The researcher decides to use Xiang’s curve to model P(a T b)

State one limitation of Xiang’s model.

TOTAL FOR STATISTICS IS 50 MARKS

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.

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.

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 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.

The variable x was measured to the nearest whole number. Forty observations are given in the table below.

A histogram was drawn and the bar representing the 10 – 15 class has a width of 2 cm and a height of 5 cm. For the 16 – 18 class find

the width,

the height

of the bar representing this class.

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 histogram in Figure 1 shows the time taken, to the nearest minute, for 140 runners to complete a fun run.

Use the histogram to calculate the number of runners who took between 78.5 and 90.5 minutes to complete the fun run.