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Review ™ stands for non-foundation part of the syllabus

2008_CE_MATH_S5_LN19e_Uses and Abuses of Statistics

LECTURE NOTES

Uses and Abuses of Statistics

CERTIFICATE MATHEMATICS

2008 Topic 19

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Topic 19

Uses and Abuses of Statistics

S.T.Cheung

CERTIFICATE MATHEMATICS LECTURE NOTES TOPIC 1

Uses and Abuses of Statistics A.

Statistical Graphs 1.

Bar Charts The bar chart represents the data by separated rectangular bars, one for each category. The height of each bar is proportional to the frequency of the corresponding category. A bar chart is useful when we want to compare the frequencies in the categories with each other.

Example 1 The following bar chart shows the distribution of means of transport to school of F.1A students. Means of transport to school of F.1A students 12 11

Number of students

10 9 8 7 6 5 4 3 2 1 0

(a) (b) (c)

Minibus

MTR

Bus Transport

Train

Others

How many students go to school by trains? How many students go to school by buses? Find the total number of students who go to school by minibuses, MTR, buses or trains.

Solution: (a) (b) (c)

2.

6 students go to school by trains. 10 students go to school by buses. Total number of students = 3 + 9 + 10 + 6 = 28

Pie Charts A pie chart is a circle divided into a number of sectors that represent the various categories. The area (or the angle at the centre) of each sector is proportional to the frequency in that category. A pie chart is most useful when we want to display the relative size of each category to the whole entity.

Example 2 The following pie chart shows the most favourite sports of 720 youngsters.

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Most favourite sports of 720 youngsters

Swimming

Badminton

90

120

38 Others

40

72 Tennis

Football

(a) (b)

How many youngsters like swimming most? Which is the most favourite sport? How many youngsters like it most?

Solution: (a)

Number of youngsters = 720 ×

90° 360°

= 180 (b)

3.

The most favourite sport is badminton. 120° Number of youngsters = 720 × 360° = 240

Line Graphs Suppose the frequency of each category is plotted above the point on the horizontal axis representing that category. The graph obtained by joining those points with line segments is called a line graph. A line graph is usually used when the categories are time units.

Example 3 The following shows the average age of Hong Kong citizens from 1961 to 1996. The average age of Hong Kong citizens from 1961 to 1996

40

Age (years old)

35

30

25

20

0 1961

1966

1971

1976

1981

1986

1991

1996

Year

(a) (b) (c)

What was the average age in 1991? In which year the average age was the lowest? What was the average age that year? Has the average age of Hong Kong citizens been increasing or decreasing since 1966?

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Solution: (a) (b) (c) 4.

The average age was 35 in 1991. The average age was the lowest in 1961. It was 25. The average age of Hong Kong citizens has been increasing since 1966.

Scatter diagram (a) (b)

Dots are used to present data. Relationship between tow sets of data can be found from the distribution of the data.

Example 4 The following shows a group of female interviewees’ age and height. Age and height

160

Height (cm)

150

140

130

120

110

0 5

(a) (b) (c) (d)

10

15 Age (years old)

What is the age of the eldest and the youngest interviewees respectively? What is the age of the tallest interviewee? Find the relationship between the age and height of female aged under 15? In general, how does the height change with the age?

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20

25

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Solution: (a) (b) (c) (d)

B.

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The eldest one is 25 years old, the youngest one is 6 years old. The tallest one is 16 years old. In general, the older the person is, the taller he/she is. For interviewees under 15 years old, the older the person is, the taller she is. But for interviewees between 15 years old and 25 years old, the height of them does not change with their age accordingly.

Method of Statistical Surveys ™ 1.

The statistical surveys basically have two kinds, which are census and sampling.

2.

Census is the inclusion of every member of the defined target population into the survey. For example, the Census and Statistics Department of Hong Kong carries out a population census every ten years to provide valuable and reliable information for the Hong Kong government and other organizations.

3.

Sampling is the selection of a comparatively small part of the population to form a sample. Then we use the information gained from the sample to infer the characteristics of the population as a whole. Population Sample

Select to form a sample from a population Sample

Investigate the sample Population Sample Inference

Make an inference about the population 4.

It is obvious that we cannot get the exact answer from sampling, but why do we use sampling and not census ? Here are the major reasons that samples are used. (a) Conducting a census may need plenty of resources, like manpower and time, which are sometimes far more than those we can afford. (b) In some cases, it is infeasible to collect data from the whole population, and thus sample surveys must be conduced. The following are some examples. i. It is infeasible to collect every bit of air in Hong Kong to investigate the degree of air pollution in Hong Kong. ii. Testing all products in a quality control process may reduce the production efficiency, and may even damage the products.

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iii.

C.

Topic 19 Uses and Abuses of Statistics

S.T.Cheung

It is infeasible to identify all drug addicts, and thus it is impossible to interview all of them for comments on community support.

Sampling Methods ™ 1.

In simple random sampling, a sample is drawn in such a way that every member of the population has an equal chance of being selected. The sample obtained is called a simple random sample. Population Sample

2.

Systematic sampling consists of taking elements at equal intervals of time or space after the first sample element is selected at random. Population

Sample 3.

In stratified random sampling, a stratified random sample is obtained by dividing the population into nonoverlapping subpopulations, called strata, and then drawing a simple random from each stratum. Population Sample

4.

In convenience sampling, a sample is selected conveniently from a part of the whole population. Therefore, convenience sampling does not involve a random selection of data from the whole population.

D. Data Collection Method ™ 1.

Recorded Information In our modern society, considerable data in various fields have been collected and compiled by organizations, researchers and companies for administration and government purposes. Hong Kong Annual Digest of Statistics published by tile Census and Statistics Department is a good source of data for planning housing, transport, education as well as social and economic researches. Information in medical records, police files and transaction records of a supermarket are useful for medical research, criminal studies, market trend and consumer behaviour analysis respectively.

2.

Direct Observation In scientific research and quality control, data are often obtained by direct observation or measurement.

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Environmental researchers collect samples of river water to measure the extent of water pollution. Quality controllers perform physical checking in production. It is also applicable to some social and marketing surveys. For example, direct observation is useful in surveying the way in which people might react when they see a $500 note in a street. Fieldworkers may be assigned to count the number of shoppers visiting a shopping mall at a certain tune period. 3.

E.

Questionnaire (a)

Face-to face interviews Face-to face interview (or called personal interview) is a common way of collecting data in opinion polls and attitude surveys in marketing. Trained interviewers meet selected persons based on certain sampling scheme, ask prepared questions and record the interviewees’ responses.

(b)

Telephone interviews As telephone is a basic house household utility now, many opinion polls and market research use the method of telephone interview to collect data. Trained interviewers contact selected sample members over the phone, ask prepared questions and record the answers.

(c)

Postal Inquiries A questionnaire is mailed to selected individuals in the sample with a request that it is completed and returned by mail to the survey organisation or person.

(d)

Internet Inquiries With tile booming of the Internet in recent years, some surveys employ Internet inquiry. An imitative ballot or a questionnaire is posted on a web site. Members of a sample or visitors of the web site are requested to vote or complete the questionnaire. After clicking the submit button, the response will be sent to the web server through the Internet and stored in it.

Abuses of Statistics ™ 1.

Observe the scales of graphs

Example 5 ™ The following two statistical diagrams are drawn to show the sales of a certain product of a company from 2002 to 2004.

Which graph is more likely to be misleading? What kind of false impression does it give? Solution: Fig. A is more likely to be misleading. The scale of the sales is not starting from zero, it gives a false impression that the sales have double itself from 2002 to 2004.

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Example 6 ™ A manager used the following statistical diagram to show the profits of ABC Company in the years 2003 and 2004.

Do you think his diagram is misleading? If yes, give your reasons and draw a suitable statistical graph to present the data. Solution: The sizes of the graphs have been distorted in such a way to mislead people that the profit of ABC Company in 2003 is 4 times that in 2004. Below is the suitable statistical graph.

2.

Note how averages or other statistical measures are calculated.

Example 7 ™ Political party X conducted a random sampling survey to ask people about their opinions on a new government policy. The results are as follows: Age group 20 – 39 40 – 59 60 – 79 (a)

Sample size 56 48 36

No. of people who favour the new policy 42 15 20

Which age group has the highest percentage of people who favour the new policy?

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(b)

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A reporter calculates the overall percentage by taking the average of the percentage of people who favour the new policy in each age group. Is the calculation correct? If not, find the true percentage.

Solution: (a)

(b)

3.

In age group 20 – 39, the percentage of people who favour the new policy 42 = × 100% = 75% 56 In age group 40 – 59, the percentage of people who favour the new policy 15 = × 100% = 31.25% 48 In age group 60 – 79, the percentage of people who favour the new policy 20 = × 100% = 55.5& % 36 ∴ The age group 20 – 39 has the highest percentage of people who favour the new policy. No. 42 + 15 + 20 × 100% The percentage of people who favour the new policy = 56 + 48 + 36 77 = × 100% 140 = 55%

Check whether the sample is representative.

Example 8 ™ The activity master of a school conducted a survey to find out the students’ performance in various clubs in the school. The following table shows some figures of the survey. Clubs Chess Club Physics Club Chemistry Club History Club Geography Club English Club (a) (b)

Number of members in the club 5 40 25 80 35 28

% of members who joined over 75% of activities held by the club in 2004 100% 70% 60% 50% 82.9% 53.6%

Judging from the percentage alone, which club’s activities was the most attractive? If we consider both the number of members and the corresponding percentages, did the club given by the answer in (a) has the most satisfactory result? Explain briefly.

Solution: (a) (b)

Chess Club No. Although the members in Chess Club were more willing to join the activities than the others, there were only 5 members in Chess Club, it is indicated that the Club was not attractive and not satisfactory.

Example 9 ™ A school assigns class number to their students alternately according to their genders. (i.e. odd numbers for girls and even numbers for boys.) The form master Mr. Fok conducts a survey to ask students, whose class numbers are multiples of 6, about their favourite ball games. (a) Would you say the data collected from the sample are representative? Why? (b) Which method of sampling would you suggest Mr. Fok to use? Solution: (a)

(b)

No. The student whose class number is a mulitple of 6 (must be even) must be a boy. So, the survey can only collect information about the boys. Randomly select 5 boys and 5 girls from each class and collect the information. 2

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CERTIFICATE MATHEMATICS MC QUESTION TOPIC 19

Uses and Abuses of Statistics [CE96/32] The bar chart below shows the number of electronic dictionaries sold in a shop last week:

Statistical Graphs [CE83/30] The pie chart shows how a boy spends the 24 hours of a day. If the boy spends 4 hours playing, how much time does he spend watching television ?

Of those electronic dictionaries sold last week, what percentage were sold on Sunday? A. B. C. D. E.

1 hour 2 hours 3 hours 4 hours 5 hours

A. B. C. D. E.

[CE94/30] In the figure, the pie chart shows the monthly expenditure of a family. If the family spends $4 800 monthly on rent, what is the monthly expenditure on entertainment?

16% 18% 20% 22.5% 25%

Frequency

[CE01/5] The bar chart below shows the distribution of scores in a test. Find the percentage of scores which are less than 3.

Score

A. B. C. D. E.

A. B. C. D. E.

$ 240 $ 600 $ 720 $ 1 800 $ 12 000

35% 40% 50% 65% 70%

[CE02/33] The pie chart below shows the expenditure of a family in January 2002. The percentage of the expenditure on Rent was

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Answers Travelling

Statistical Graphs Saving

Clothing

[CE83/30] [CE94/30] [CE96/32] [CE01/5] [CE02/33] [CE04/35]

Rent Food

A. B. C. D.

12.5%. 22.5%. 25%. 45%.

[CE04/35] The pie chart below shows the expenditure of a student in March 2004. If the student spent $520 on meals, then the student’s total expenditure on entertainment and clothing was

A. B. C. D.

$780. $1092. $1352. $1872.

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C B C B A A