2.01c Sampling techniques: simple random, opportunity, etc

1. (a) State one disadvantage of using quota sampling compared with simple random sampling.

In a university 8\% of students are members of the university dance club.
A random sample of 36 students is taken from the university.
The random variable \(X\) represents the number of these students who are members of the dance club.
(b) Using a suitable model for \(X\), find

1. \(\mathrm { P } ( X = 4 )\)
2. \(\mathrm { P } ( X \geqslant 7 )\) Only 40\% of the university dance club members can dance the tango.
   (c) Find the probability that a student is a member of the university dance club and can dance the tango. A random sample of 50 students is taken from the university.
   (d) Find the probability that fewer than 3 of these students are members of the university dance club and can dance the tango. \section*{Question 1 continued.}

5 Ali collected data from a random sample of 200 workers and recorded the number of days they each worked from home in the second week of September 2019. These data are shown in Fig. 5.1. \begin{table}[h] \end{table}

1. Represent the data by a suitable diagram.
2. Calculate
   Ali then collected data from a different random sample of 200 workers for the same week in September 2019. The mean number of days worked from home for this sample was 1.94 and the standard deviation was 1.75.
3. Explain whether there is any evidence to suggest that one or both of the samples must be flawed. Fig. 5.2 shows a cumulative frequency diagram for the ages of the workers in the first sample who worked from home on at least one day. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e0b502a8-c742-4d78-993c-8c0c7329ec9c-04_671_1362_1452_241} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
   \end{figure} Ali concludes that \(90 \%\) of the workers in this sample who worked from home on at least one day were under 60 years of age
4. Explain whether Ali's conclusion is correct.

6 The pre-release material contains information about employment rates in London boroughs. The graph shows employment rates for Westminster between 2006 and 2019. \begin{figure}[h] \end{figure} A local politician stated that the diagram shows that more than \(60 \%\) of seventy-year-olds were in employment throughout the period from 2006 to 2019.

1. Use your knowledge of the pre-release material to explain whether there is any evidence to support this statement. In order to estimate the employment rate in 2020, two different models were proposed using the LINEST function in a spreadsheet. Model 1 (using all the data from 2006 onwards) \(\mathrm { Y } = 0.549 \mathrm { x } - 1040\), Model 2 (using data from 2017 onwards) \(\mathrm { Y } = 2.65 \mathrm { x } - 5280\),
   where \(Y =\) employment rate and \(x =\) calendar year. It was subsequently found that the employment rate in Westminster in 2020 was 68.4\%.
2. Determine which of the two models provided the better estimate for the employment rate in Westminster in 2020.
3. Use your knowledge of the pre-release material to explain whether it would be appropriate to use either model to estimate the employment rate in 2020 in other London boroughs.
4. What does model 2 predict for employment rates in Westminster in the long term?

11 The pre-release material contains information about the Median Income of Taxpayers and the Percentage of Pupils Achieving at Least 5 A*- C grades, including English and Maths, at the end of KS4 in different areas of London. Alex is investigating whether there is a relationship between median income and the percentage of pupils achieving at least 5 A* - C grades, including English and Maths, at the end of KS4. Alex decides to use the first 12 rows of data for 2014-5 from the pre-release data as a sample. The sample is shown in Fig. 11.1. \begin{table}[h] \end{table}

1. Explain whether the data in Fig. 11.1 is a simple random sample of the data for 2014-5.
2. The City of London is included in Alex's sample. Explain why Alex is not able to use the data for the City of London in this investigation. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 11.2 shows a scatter diagram showing Percentage of Pupils against Median Income for all of the areas of London for which data is available.} \includegraphics[alt={},max width=\textwidth]{e0b502a8-c742-4d78-993c-8c0c7329ec9c-09_716_1378_356_244}
   \end{figure} Fig. 11.2 Alex identifies some outliers.
3. On the copy of Fig. 11.2 in the Printed Answer Booklet, ring three of these outliers. Alex then discards all the outliers and uses the LINEST function on a spreadsheet to obtain the following model. \(\mathrm { P } = 0.0009049 \mathrm { M } + 37.38\),
   where \(P =\) percentage of pupils and \(M =\) median income.
4. Show that the model is a good fit for the data for Hackney.
5. Use the model to find an estimate of the value of \(P\) for City of London.
6. Give two reasons why this estimate may not be reliable. Alex states that more than 50\% of the pupils in London achieved at least a grade C at the end of KS4 in English and Maths in 2014-5.
7. Use the information in Fig. 11.2 together with your knowledge of the pre-release material to explain whether there is evidence to support this statement.

12 Doctors are investigating the weights of adult males registered at their surgery. One week they collect a sample by noting the weight in kilograms of all the adult males who have an appointment at their surgery.

1. State the sampling method they use.
2. Explain why this method will not generate a simple random sample of all the adult males registered at their surgery. They represent the data using a histogram. \includegraphics[max width=\textwidth, alt={}, center]{82438df0-6550-4ffd-92d8-3c67bec59a6b-09_1166_1243_726_233} An incomplete frequency table for the data is shown below.

   Weight in kg	\(50 -\)	\(65 -\)	\(75 -\)	\(80 -\)	\(90 -\)	\(100 - 120\)
   Frequency		8

3. Complete the copy of the frequency table in the Printed Answer Booklet. One of these patients is selected at random.
4. Determine an estimate of the probability that he weighs either less than 60 kg or more than 110 kg .
5. Explain why your answer to part (d) is an estimate and not exact.

3 A researcher is conducting an investigation into the number of portions of fruit adults consume each day. The researcher decides to ask 50 men and 50 women to complete a simple questionnaire.

1. State the type of sampling procedure the researcher is using.
2. Write down one disadvantage of this sampling procedure. The researcher represents the data in Fig. 3.1. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Number of portions of fruit consumed by adults} \includegraphics[alt={},max width=\textwidth]{c08a2212-3104-425e-8aee-7f2d46f23924-06_531_991_701_248}
   \end{figure} Fig. 3.1
3. Describe the shape of the distribution. The data are summarised in the frequency table in Fig. 3.2. \begin{table}[h]

   Number of portions of fruit	0	1	2	3	4	5
   Number of adults	18	34	26	11	7	4

   \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{table}
4. For the data in Fig. 3.2, use your calculator to find
   Give your answers correct to 2 decimal places. A second researcher chooses a proportional stratified sample of 100 children from years 5 and 6 in a certain primary school. There are 220 children to choose from. In year 5 there are 125 children, of whom 81 are boys.
5. How many year 5 girls should be included in the sample? The second researcher found that the mean number of portions of fruit consumed per day by the children in this sample was 1.61 and the standard deviation was 0.53 .
6. Comment on the amount of fruit consumed per day by the children compared to the amount of fruit consumed per day by the adults.

11 James is investigating the amount of time retired people spend each day using social media. He collects a sample by advertising in a local newspaper for people to complete an online survey.

1. State
   James processes his data in order to draw a histogram. His table of results is shown below.

   Time spent using social media in minutes	\(0 -\)	\(15 -\)	\(30 -\)	\(60 -\)	\(120 - 240\)
   Number of people per minute	12.2	14.0	8.4	7.3	3.1

2. Show that the size of the sample is 1455 .
3. Calculate an estimate of the probability that a retired person spends more than an hour per day using social media.

7 A farmer has 200 apple trees. She is investigating the masses of the crops of apples from individual trees. She decides to select a sample of these trees and find the mass of the crop for each tree.

1. Explain how she can select a random sample of 10 different trees from the 200 trees. The masses of the crops from the 10 trees, measured in kg, are recorded as follows. \(\begin{array} { l l l l l l l l l l } 23.5 & 27.4 & 26.2 & 29.0 & 25.1 & 27.4 & 26.2 & 28.3 & 38.1 & 24.9 \end{array}\)
2. For these data find

8 A garden centre stocks coniferous hedging plants. These are displayed in 10 rows, each of 120 plants. An employee collects a sample of the heights of these plants by recording the height of each plant on the front row of the display.

1. Explain whether the data collected by the employee is a simple random sample. The data are shown in the cumulative frequency curve below. \includegraphics[max width=\textwidth, alt={}, center]{11788aaf-98fb-4a78-8a40-a40743b1fe15-06_1376_1344_680_233} The owner states that at least \(75 \%\) of the plants are between 40 cm and 80 cm tall.
2. Show that the data collected by the employee supports this statement.
3. Explain whether all samples of 120 plants would necessarily support the owner's statement.

9 A teacher is investigating how pupils travel to and from school each day. Pupils can either travel by bus, train, car, bicycle or walk. The teacher decides to collect a sample of size 60 for the investigation.

1. The teacher lives in a village 10 miles away from the school. Explain how collecting a sample which just consists of pupils who live in the same village as the teacher might introduce bias. The table below shows how many students there are in each year.

   Year 7	Year 8	Year 9	Year 10	Year 11
   86	105	107	101	101

2. The teacher decides to use the method of proportional stratified sampling. Calculate the number of pupils in the sample who are in Year 9. The teacher generates a sample of 10 pupils from the 86 in Year 7 by listing them in alphabetical order and selecting the first name on the list and every ninth name thereafter.
3. Explain whether this method will generate a simple random sample of the pupils who travel in Year 7.

8 Rosella is carrying out an investigation into the age at which adults retire from work in the city where she lives. She collects a sample of size 50 , ensuring this comprises of 25 randomly selected retired men and 25 randomly selected retired women.

1. State the name of the sampling method she uses. Fig. 8.1 shows the data she obtains in a frequency table and Fig. 8.2 shows these data displayed in a histogram. \begin{table}[h]

   Age in years at retirement	\(45 -\)	\(50 -\)	\(55 -\)	\(60 -\)	\(65 -\)	\(70 -\)	\(75 - 80\)
   Frequency density	0.4	1.8	2.4	2.2	1.8	1.2	0.2

   \captionsetup{labelformat=empty} \caption{Fig. 8.1}
   \end{table} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-08_805_1006_1160_244} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
   \end{figure}
2. How many people in the sample are aged between 50 and 55? Rosella obtains a list of the names of all 4960 people who have retired in the city during the previous month.
3. Describe how Rosella could collect a sample of size 200 from her list using
   Rosella collects two simple random samples, one of size 200 and one of size 500, from her list. The histograms in Fig. 8.3 show the data from the sample of size 200 on the left and the data from the sample of 500 on the right. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-09_659_1909_388_77} \captionsetup{labelformat=empty} \caption{Fig. 8.3}
   \end{figure}
4. With reference to the histograms shown in Fig. 8.2 and Fig. 8.3, explain why it appears reasonable to model the age of retirement in this city using the Normal distribution. Summary statistics for the sample of 500 are shown in Fig. 8.4. \begin{table}[h]

   Statistics
   n	500
   Mean	60.0515
   \(\sigma\)	6.5717
   s	6.5783
   \(\Sigma x\)	30025.7601
   \(\Sigma \mathrm { x } ^ { 2 }\)	1824686.322
   Min	36.0793
   Q1	55.2573
   Median	59.9202
   Q3	64.4239
   Max	81.742

   \captionsetup{labelformat=empty} \caption{Fig. 8.4}
   \end{table}
5. Use an appropriate Normal model based on the information in Fig. 8.4 to estimate the number of people aged over 65 who retired in the city in the previous month.
6. Identify a limitation in using this model to predict the number of people aged over 65 retiring in the following month.

9 A company supplies computers to businesses. In the past the company has found that computers are kept by businesses for a mean time of 5 years before being replaced. Claud, the manager of the company, thinks that the mean time before replacing computers is now different.

1. Describe how Claud could obtain a cluster sample of 120 computers used by businesses the company supplies. Claud decides to conduct a hypothesis test at the \(5 \%\) level to test whether there is evidence to suggest that the mean time that businesses keep computers is not 5 years. He takes a random sample of 120 computers. Summary statistics for the length of time computers in this sample are kept are shown in Fig. 9. \begin{table}[h]

   Statistics
   \(n\)	120
   Mean	4.8855
   \(\sigma\)	2.6941
   \(s\)	2.7054
   \(\Sigma x\)	586.2566
   \(\Sigma x ^ { 2 }\)	3735.1475
   Min	0.1213
   Q1	2.5472
   Median	4.8692
   Q3	7.0349
   Max	9.9856

   \captionsetup{labelformat=empty} \caption{Fig. 9}
   \end{table} \section*{(b) In this question you must show detailed reasoning.}
   • State the hypotheses for this test, explaining why the alternative hypothesis takes the form it does.
   • Use a suitable distribution to carry out the test.

10 Ben has an interest in birdwatching. For many years he has identified, at the start of the year, 32 days on which he will spend an hour counting the number of birds he sees in his garden. He divides the year into four using the Meteorological Office definition of seasons. Each year he uses stratified sampling to identify the 32 days on which he will count the birds in his garden, drawn equally from the four seasons. Ben's data for 2019 are shown in the stem and leaf diagram in Fig. 10.1. \begin{table}[h] \end{table}

1. Suggest a reason why Ben chose to use stratified sampling instead of simple random sampling.
2. Describe the shape of the distribution.
3. Explain why the mode is not a useful measure of central tendency in this case.
4. For Ben's sample, determine
   Ben found a boxplot for the sample of size 32 he collected using stratified sampling in 2015. The boxplot is shown in Fig. 10.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c9d14a4d-a1c8-42ad-9c0b-42cef6b3612f-06_483_1163_1982_242} \captionsetup{labelformat=empty} \caption{Fig. 10.2}
   \end{figure} In 2016 Ben replaced his hedge with a garden fence.
   Ben now believes that
   Jane says she can tell that the data for 2015 is definitely uniformly distributed by looking at the boxplot.
5. Explain why Jane is wrong.

2. Bill owns a restaurant. Over the next four weeks Bill decides to carry out a sample survey to obtain the customers' opinions.

1. Suggest a suitable sampling frame for the sample survey.
2. Identify the sampling units.
3. Give one advantage and one disadvantage of taking a census rather than a sample survey. Bill believes that only \(30 \%\) of customers would like a greater choice on the menu. He takes a random sample of 50 customers and finds that 20 of them would like a greater choice on the menu.
4. Test, at the \(5 \%\) significance level, whether or not the percentage of customers who would like a greater choice on the menu is more than Bill believes. State your hypotheses clearly.

1. (i) (a) State the conditions under which the Poisson distribution may be used as an approximation to the binomial distribution.

A factory produces tyres for bicycles and \(0.25 \%\) of the tyres produced are defective. A company orders 3000 tyres from the factory.
(b) Find, using a Poisson approximation, the probability that there are more than 7 defective tyres in the company's order.
(ii) At the company \(40 \%\) of employees are known to cycle to work. A random sample of 150 employees is taken. The random variable \(C\) represents the number of employees in the sample who cycle to work.
(a) Describe a suitable sampling frame that can be used to take this sample.
(b) Explain what you understand by the sampling distribution of \(C\) Louis uses a normal approximation to calculate the probability that at most \(\alpha\) employees in the sample cycle to work. He forgets to use a continuity correction and obtains the incorrect probability 0.0668 Find, showing all stages of your working,
(c) the value of \(\alpha\) (d) the correct probability.

6. The owner of a very large youth club has designed a new method for allocating people to teams. Before introducing the method he decided to find out how the members of the youth club might react.

1. Explain why the owner decided to take a random sample of the youth club members rather than ask all the youth club members.
2. Suggest a suitable sampling frame.
3. Identify the sampling units. The new method uses a bag containing a large number of balls. Each ball is numbered either 20, 50 or 70
   When a ball is selected at random, the random variable \(X\) represents the number on the ball where $$\mathrm { P } ( X = 20 ) = p \quad \mathrm { P } ( X = 50 ) = q \quad \mathrm { P } ( X = 70 ) = r$$ A youth club member takes a ball from the bag, records its number and replaces it in the bag. He then takes a second ball from the bag, records its number and replaces it in the bag. The random variable \(M\) is the mean of the 2 numbers recorded. Given that $$\mathrm { P } ( M = 20 ) = \frac { 25 } { 64 } \quad \mathrm { P } ( M = 60 ) = \frac { 1 } { 16 } \quad \text { and } \quad q > r$$
4. show that \(\mathrm { P } ( M = 50 ) = \frac { 1 } { 16 }\)

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1. (a) State one characteristic of a population that would make a census a practical alternative to sampling.

A leisure centre has 2500 members.
It asks a sample of 300 members for their opinions on the fees it charges for using the centre. For the sample,
(b) (i) identify a suitable sampling frame,
(ii) identify a sampling unit. The leisure centre has the following pieces of information. \(A\) is the list of the different types of membership that can be paid for by members. \(B\) is the mean of the membership fees paid by all 2500 members. \(C\) is the number in the sample of 300 members who are satisfied with the fees they pay.
(c) State the piece of information that is a statistic. Give a reason for your answer.

3 Jian owns a large group of shops. She decides to visit a random sample of the shops to check if the stocktaking system is being used incorrectly.

1. Suggest a suitable sampling frame for Jian to use.
2. Identify the sampling units.
3. Give one advantage and one disadvantage of taking a sample rather than a census. Jian believes that the stocktaking system is being used incorrectly in \(40 \%\) of the shops.
   To investigate her belief, a random sample of 30 of the shops is taken.
4. Using a 5\% level of significance, find the critical region for a two-tailed test of Jian's belief.
   You should state the probability in each tail, which should each be as close as possible to 2.5\% The total number of shops, in the sample of 30, where the stocktaking system is being used incorrectly is 20
5. Using the critical region from part (d), state what this suggests about Jian's belief. Give a reason for your answer. Jian introduces a new, simpler, stocktaking system to all the shops.
   She takes a random sample of 150 shops and finds that in 47 of these shops the new stocktaking system is being used incorrectly.
6. Using a suitable approximation, test, at the \(5 \%\) level of significance, whether or not there is evidence that the proportion of shops where the stocktaking system is being used incorrectly is now less than 0.4 You should state your hypotheses and show your working clearly.

6. A magazine has a large number of subscribers who each pay a membership fee that is due on January 1st each year. Not all subscribers pay their fee by the due date. Based on correspondence from the subscribers, the editor of the magazine believes that \(40 \%\) of subscribers wish to change the name of the magazine. Before making this change the editor decides to carry out a sample survey to obtain the opinions of the subscribers. He uses only those members who have paid their fee on time.

1. Define the population associated with the magazine.
2. Suggest a suitable sampling frame for the survey.
3. Identify the sampling units.
4. Give one advantage and one disadvantage that would have resulted from the editor using a census rather than a sample survey. As a pilot study the editor took a random sample of 25 subscribers.
5. Assuming that the editor's belief is correct, find the probability that exactly 10 of these subscribers agreed with changing the name. In fact only 6 subscribers agreed to the name being changed.
6. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not the percentage agreeing to the change is less that the editor believes. The full survey is to be carried out using 200 randomly chosen subscribers.
7. Again assuming the editor's belief to be correct and using a suitable approximation, find the probability that in this sample there will be least 71 but fewer than 83 subscribers who agree to the name being changed. \section*{END}

1. (a) Explain what you understand by a census.

Each cooker produced at GT Engineering is stamped with a unique serial number. GT Engineering produces cookers in batches of 2000. Before selling them, they test a random sample of 5 to see what electric current overload they will take before breaking down.
(b) Give one reason, other than to save time and cost, why a sample is taken rather than a census.
(c) Suggest a suitable sampling frame from which to obtain this sample.
(d) Identify the sampling units.

4. Explain what you understand by

1. a sampling unit,
2. a sampling frame,
3. a sampling distribution.

3. An athletics teacher has kept careful records over the past 20 years of results from school sports days. There are always 10 competitors in the javelin competition. Each competitor is allowed 3 attempts and the teacher has a record of the distances thrown by each competitor at each attempt. The random variable \(D\) represents the greatest distance thrown by each competitor and the random variable \(A\) represents the number of the attempt in which the competitor achieved their greatest distance.

1. State which of the two random variables \(D\) or \(A\) is continuous. A new athletics coach wishes to take a random sample of the records of 36 javelin competitors.
2. Specify a suitable sampling frame and explain how such a sample could be taken.
   (2 marks)
   The coach assumes that \(\mathrm { P } ( A = 2 ) = \frac { 1 } { 3 }\), and is therefore surprised to find that 20 of the 36 competitors in the sample achieved their greatest distance on their second attempt. Using a suitable approximation, and assuming that \(\mathrm { P } ( A = 2 ) = \frac { 1 } { 3 }\),
3. find the probability that at least 20 of the competitors achieved their greatest distance on their second attempt.
   (6 marks)
4. Comment on the assumption that \(\mathrm { P } ( A = 2 ) = \frac { 1 } { 3 }\).

1. A journalist is going to interview a sample of 10 players from the 60 players in a local football club. The journalist uses the random numbers on page 27 of the formula booklet and starts at the top of the 10th column, where the first number is 96

The journalist worked down the 10th column to select 10 numbers. The first 3 numbers selected were: 33, 15 and 23

1. Find the other 7 numbers to complete the sample of ten. There are 24 girls and 36 boys who play football for the club.
   The journalist labels the girls from 1 to 24 and the boys from 25 to 60
2. Show how the journalist can use her 10 random numbers to select a stratified sample of 10 players from the club to interview. The club provided the journalist with a list of the players in ascending order of ages, numbered 1 to 60. The journalist uses the 10 random numbers to select a simple random sample of the players.
3. State, giving a reason, a group of players who may not be represented in this sample.

1. The Headteacher of a school is thinking about making changes to the school day. She wants to take a sample of 60 students so that she can find out what the students think about the proposed changes.

The names of the 1200 students of the school are listed alphabetically.

1. Explain how the Headteacher could take a systematic sample of 60 students.
2. 1. Explain why systematic sampling is likely to be quicker than simple random sampling in this situation.
   2. With reference to this situation,
      • explain why systematic sampling may introduce bias compared to simple random sampling
3. give an example of the bias that may occur when using this alphabetical list
When the Headteacher completes the systematic sample of size 60 she finds that 6 students were to be selected from Year 9. The Head of Mathematics suggests that a stratified sample of size 60 would be a more appropriate method. There were 200 students in Year 9.
4. Explain why this suggests that a stratified sample of size 60 may be better than the systematic sample taken by the Headteacher.