## A small company produces cylindrical wooden pegs for making garden chairs. The

A small company produces cylindrical wooden pegs for making garden chairs. The lengths and diameters of the 242 pegs produced yesterday have been measured independently by two employees, and their results are given in the following table.

a. On the same axes, draw two cumulative frequency graphs: one for lengths and one for diameters.

b. Correct to the nearest millimetre, the lengths and diameters of n of these pegs are equal. Find the least and greatest possible value of n.

c. A peg is acceptable for use when it satisfies both l ≥ 2.8 and d < 2.2. Explain why you cannot obtain from your graphs an accurate estimate of the number of these 242 pegs that are acceptable. Suggest what the company could do differently so that an accurate estimate of the proportion of acceptable pegs could be obtained.

## The following table shows the ages of the students currently at a

The following table shows the ages of the students currently at a university, given by percentage. Ages are rounded down to the number of whole years.

a. Represent the data in a percentage cumulative frequency polygon.

b. The oldest 8% of these students qualify as ‘mature’. Use your polygon to estimate the minimum age requirement for a student to be considered mature. Give your answer to the nearest month.

c. Of the 324 students who are 18–19 years old, 54 are not expected to find employment within 3 months of finishing their course.

i. Calculate an estimate of the number of current students who are expected to find employment within 3 months of finishing their course.

ii. What assumptions must be made to justify your calculations in part c i? Are these assumptions reasonable? Do you expect your estimate to be an overestimate or an underestimate?

## The thicknesses, kmm, of some steel sheets are represented in the histogram.

The thicknesses, kmm, of some steel sheets are represented in the histogram. It is given that k < 0.4 for 180 sheets.

a. Find the ratio between the frequencies of the three classes. Give your answer in simplified form.

b. Find the value of n, given that frequency density measures sheets per nmm.

c. Calculate an estimate of the number of sheets for which:

i. k < 0.5

ii. 0.75 ≤ k < 0.94.

d. The sheets are classified as thin, medium or thick in the ratio 1: 3 :1.

Estimate the thickness of a medium sheet, giving your answer in the form a

## A fashion company selected 100 12-year-old boys and 100 12-year-old girls to

A fashion company selected 100 12-year-old boys and 100 12-year-old girls to audition as models. The heights, hcm, of the selected children are represented in the following graph.

a. What features of the data suggest that the children were not selected at random?

b. Estimate the number of girls who are taller than the shortest 50 boys.

c. What is the significance of the value of h where the graphs intersect?

d. The shortest 75 boys and tallest 75 girls were recalled for a second audition. On a cumulative frequency graph, show the heights of the children who were not recalled.

## A histogram is drawn with three columns whose widths are in the

A histogram is drawn with three columns whose widths are in the ratio 1: 2 : 4. The frequency densities of these classes are in the ratio 16 :12 : 3, respectively.

a. Given that the total frequency of the data is 390, find the frequency of each class. [3]

b. The classes with the two highest frequencies are to be merged and a new histogram drawn. Given that the height of the column representing the merged classes is to be 30cm, find the correct height for the remaining column.

c. Explain what problems you would encounter if asked to construct a histogram in which the classes with the two lowest frequencies are to be merged.

## The histograms below illustrate the number of hours of sunshine during August

The histograms below illustrate the number of hours of sunshine during August in two regions, A and B. Neither region had more than 8 hours of sunshine per day.

a. Explain how you know that some information for one of the regions has been omitted.

b. After studying the histograms, two students make the following statements.

● Bindu: There was more sunshine in region A than in region B during the first 2 weeks of August.

● Janet: In August there was less sunshine in region A than in region B.

Discuss these statements and decide whether or not you agree with each of them. In each case, explain your reasoning.

## The reaction times, t seconds, of 66 participants were measured in an

The reaction times, t seconds, of 66 participants were measured in an experiment and presented below.

Time (t seconds) ………………….              No. participants (cf )
t < 1.5 …………………………………………………………. 0
t < 3.0 …………………………………………………………. 3
t < 4.5 …………………………………………………………. 8
t < 6.5 ……………………………………………………….. 32
t < 8.5 ………………………………………………………. 54
t < 11.0 …………………………………………………….. 62
t < 13.0 …………………………………………………….. 66

a. Draw a cumulative frequency polygon to represent the data.

b. Use your graph to estimate:

i. The number of participants with reaction times between 5.5 and 7.5 seconds

ii. The lower boundary of the slowest 20 reaction times.

## Measurements of the distances, xmm, between two moving parts inside car engines

Measurements of the distances, xmm, between two moving parts inside car engines were recorded and are summarised in the following table. There were 156 engines of type A and 156 engines of type B.

a. Draw and label two cumulative frequency curves on the same axes.

b. Use your graphs to estimate:

i. The number of engines of each type with measurements between 0.30 and 0.70mm

ii. The total number of engines with measurements that were less than 0.55mm.

c. Both types of engine must be repaired if the distance between these moving parts is more than a certain fixed amount. Given that 16 type A engines need repairing, estimate the number of type B engines that need repairing.

## Over a 14-day period, data were collected on the number of passengers

Over a 14-day period, data were collected on the number of passengers travelling on two ferries, A and Z. The results are presented to the right.

a. How many more passengers travelled on ferry Z than on ferry A?

b. The cost of a trip on ferry A is \$12.50 and the cost of a trip on ferry Z is \$x. The takings on ferry Z were \$3.30 less than the takings on ferry A over this period. Find the value of x.

c. Find the least and greatest possible number of days on which the two ferries could have carried exactly the same number of passengers.

## Boxes of floor tiles are to be offered for sale at a

Boxes of floor tiles are to be offered for sale at a special price of \$75. The boxes claim to contain at least 100 tiles each.

a. Why would it be preferable to use a stem-and-leaf diagram rather than a bar chart to represent the numbers of tiles, which are 112, 116, 107, 108, 121, 104, 111, 106, 105 and 110?

b. How may the seller benefit if the numbers 12, 16, 7, 8, 21, 4, 11, 6, 5 and 10 are used to draw the stem-and leaf diagram instead of the actual numbers of tiles?