2.01c Sampling techniques: simple random, opportunity, etc

7 A market researcher is investigating the length of time that customers spend at an information desk. He plans to choose a sample of 50 customers on a particular day.

1. He considers choosing the first 50 customers who visit the information desk. Explain why this method is unsuitable.
   The actual lengths of time, in minutes, that customers spend at the information desk may be assumed to have mean \(\mu\) and variance 4.8. The researcher knows that in the past the value of \(\mu\) was 6.0. He wishes to test, at the \(2 \%\) significance level, whether this is still true. He chooses a random sample of 50 customers and notes how long they each spend at the information desk.
2. State the probability of making a Type I error and explain what is meant by a Type I error in this context.
3. Given that the mean time spent at the information desk by the 50 customers is 6.8 minutes, carry out the test.
4. Give a reason why it was necessary to use the Central Limit theorem in your answer to part (c).
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1. A javelin thrower noted the lengths of a random sample of 50 of her throws. The sample mean was 72.3 m and an unbiased estimate of the population variance was \(64.3 \mathrm {~m} ^ { 2 }\). Find a \(92 \%\) confidence interval for the population mean length of throws by this athlete.
2. A discus thrower wishes to calculate a \(92 \%\) confidence interval for the population mean length of his throws. He bases his calculation on his first 50 throws in a week. Comment on this method.

2 A club has 264 members, numbered from 1 to 264 . Donash wants to choose a random sample of members for a survey. In order to choose the members for the sample he uses his calculator to generate random digits. His first 20 random digits are as follows. $$\begin{array} { l l l l } 10612 & 11801 & 21473 & 22759 \end{array}$$

1. The numbers of the first two members in the sample are 106 and 121. Write down the numbers of the next two members in the sample.
2. To obtain the numbers for members after the 4th member, Donash starts with the second random digit, 0 , and obtains the numbers 061 and 211. Explain why this method will not produce a random sample.

2 Henri wants to choose a random sample from the 804 students at his college. He numbers the students from 1 to 804 and then uses random numbers generated by his calculator. The first 20 random digits produced by his calculator are as follows. $$\begin{array} { l l l l l l l l l l l l l l l l l l l l } 5 & 6 & 7 & 1 & 0 & 9 & 8 & 4 & 3 & 1 & 0 & 9 & 6 & 6 & 5 & 0 & 2 & 1 & 7 & 6 \end{array}$$ Henri's first two student numbers are 567 and 109.

1. Use Henri's digits to find the numbers of the next two students in the sample.
   There were 30 students in Henri's sample. He asked each of them how much time, \(X\) hours, they spent on social media each week, on average. He summarised the results as follows. $$n = 30 \quad \Sigma x = 610 \quad \Sigma x ^ { 2 } = 12405$$
2. Use this information to calculate an unbiased estimate of the mean of \(X\) and show that an unbiased estimate of the variance of \(X\) is less than 0.1 .
3. Henri's friend claims that Henri has probably made a mistake in his calculation of \(\Sigma x\) or \(\Sigma x ^ { 2 }\). Use your answer to part (b) to comment on this claim.

7 A researcher is investigating the actual lengths of time that patients spend with the doctor at their appointments. He plans to choose a sample of 12 appointments on a particular day.

1. Which of the following methods is preferable, and why?
   • Choose the first 12 appointments of the day.
   • Choose 12 appointments evenly spaced throughout the day.
   Appointments are scheduled to last 10 minutes. The actual lengths of time, in minutes, that patients spend with the doctor may be assumed to have a normal distribution with mean \(\mu\) and standard deviation 3.4. The researcher suspects that the actual time spent is more than 10 minutes on average. To test this suspicion, he recorded the actual times spent for a random sample of 12 appointments and carried out a hypothesis test at the 1\% significance level.
2. State the probability of making a Type I error and explain what is meant by a Type I error in this context.
3. Given that the total length of time spent for the 12 appointments was 147 minutes, carry out the test.
4. Give a reason why the Central Limit theorem was not needed in part (iii).

1 Jyothi wishes to choose a representative sample of 5 students from the 82 members of her school year.

1. She considers going into the canteen and choosing a table with five students from her year sitting at it, and using these five people as her sample. Give two reasons why this method is unsatisfactory.
2. Jyothi decides to use another method. She numbers all the students in her year from 1 to 82 . Then she uses her calculator and generates the following random numbers. $$231492 \quad 762305 \quad 346280$$ From these numbers, she obtains the student numbers \(23,14,76,5,34\) and 62 . Explain how Jyothi obtained these student numbers from the list of random numbers.

1 A residents' association has 654 members, numbered from 1 to 654 . The secretary wishes to send a questionnaire to a random sample of members. In order to choose the members for the sample she uses a table of random numbers. The first line in the table is as follows. $$\begin{array} { l l l l l l } 1096 & 4357 & 3765 & 0431 & 0928 & 9264 \end{array}$$ The numbers of the first two members in the sample are 109 and 643. Find the numbers of the next three members in the sample.

2 Amy has to choose a random sample from the 265 students in her year at college. She numbers the students from 1 to 265 and then uses random numbers generated by her calculator. The first two random numbers produced by her calculator are 0.213165448 and 0.073165196 .

1. Use these figures to find the numbers of the first four students in her sample.
   There were 25 students in Amy's sample. She asked each of them how much money, \(\\) x$, they earned in a week, on average. Her results are summarised below. $$n = 25 \quad \Sigma x = 510 \quad \Sigma x ^ { 2 } = 13225$$
2. Find unbiased estimates of the population mean and variance.
3. Explain briefly what is meant by 'population' in this question.

2 Andy and Jessica are doing a survey about musical preferences. They plan to choose a representative sample of six students from the 256 students at their college.

1. Andy suggests that they go to the music building during the lunch hour and choose six students at random from the students who are there. Give a reason why this method is unsatisfactory.
2. Jessica decides to use another method. She numbers all the students in the college from 1 to 256. Then she uses her calculator and generates the following random numbers. $$\begin{array} { l l l l l } 204393 & 162007 & 204028 & 587119 & 207395 \end{array}$$ From these numbers, she obtains six student numbers. The first three of her student numbers are 204, 162 and 7. Continue Jessica's method to obtain the next three student numbers.

2 Describe briefly how to use random numbers to choose a sample of 10 students from a year-group of 276 students.

2 Jenny has to do a statistics project at school on how much pocket money, in dollars, is received by students in her year group. She plans to take a sample of 7 students from her year group, which contains 122 students.

1. Give a suitable method of taking this sample. Her sample gives the following results. $$\begin{array} { l l l l l l l } 13.40 & 10.60 & 26.50 & 20.00 & 14.50 & 15.00 & 16.50 \end{array}$$
2. Find unbiased estimates of the population mean and variance.
3. Is the estimated population variance more than, less than or the same as the sample variance?
4. Describe what you understand by 'population' in this question.

1 A magazine conducted a survey about the sleeping time of adults. A random sample of 12 adults was chosen from the adults travelling to work on a train.

1. Give a reason why this is an unsatisfactory sample for the purposes of the survey.
2. State a population for which this sample would be satisfactory. A satisfactory sample of 12 adults gave numbers of hours of sleep as shown below. \(4.6 \quad 6.8\) 5.2
   6.2
   5.7 \(\begin{array} { l l } 7.1 & 6.3 \end{array}\) 5.6
   7.0 \(5.8 \quad 6.5\) 7.2
3. Calculate unbiased estimates of the mean and variance of the sleeping times of adults.

6 Ramesh plans to carry out a survey in order to find out what adults in his town think about local sports facilities. He chooses a random sample from the adult members of a tennis club and gives each of them a questionnaire.

1. Give a reason why this will not result in Ramesh having a random sample of adults who live in the town.
2. Describe briefly a valid method that Ramesh could use to choose a random sample of adults in the town.
   Ramesh now uses a valid method to choose a random sample of 350 adults from the town. He finds that 47 adults think that the local sports facilities are good.
3. Calculate an approximate \(90 \%\) confidence interval for the proportion of all adults in the town who think that the local sports facilities are good.
4. Ramesh calculates a confidence interval whose width is 1.25 times the width of this \(90 \%\) confidence interval. Ramesh's new interval is an \(x \%\) confidence interval. Find the value of \(x\).
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3 Luis has to choose one person at random from four people, \(A , B , C\) and \(D\). He throws a fair six-sided die. If the score is 1 , he will choose \(A\). If the score is 2 he will choose \(B\). If the score is 3 , he will choose \(C\). If the score is 4 or more he will choose \(D\).

1. Explain why the choice made by this method is not random.
2. Describe how Luis could use a single throw of the die to make a random choice.
   On another day, Luis has to choose two people at random from the same four people, \(A , B , C\) and \(D\).
3. List the possible choices of two people and hence describe how Luis could use a single throw of the die to make this random choice.

7 In a research laboratory where plants are studied, the probability of a certain type of plant surviving was 0.35 . The laboratory manager changed the growing conditions and wished to test whether the probability of a plant surviving had increased.

1. The plants were grown in rows, and when the manager requested a random sample of 8 plants to be taken, the technician took all 8 plants from the front row. Explain what was wrong with the technician's sample.
2. A suitable sample of 8 plants was taken and 4 of these 8 plants survived. State whether the manager's test is one-tailed or two-tailed and also state the null and alternative hypotheses. Using a \(5 \%\) significance level, find the critical region and carry out the test.
3. State the meaning of a Type II error in the context of the test in part (ii).
4. Find the probability of a Type II error for the test in part (ii) if the probability of a plant surviving is now 0.4.

1 Alan wishes to choose one child at random from the eleven children in his music class. The children are numbered \(2,3,4\), and so on, up to 12 . Alan then throws two fair dice, each numbered from 1 to 6 , and chooses the child whose number is the sum of the scores on the two dice.

1. Explain why this is an unsatisfactory method of choosing a child.
2. Describe briefly a satisfactory method of choosing a child.

1 There are 18 people in Millie's class. To choose a person at random she numbers the people in the class from 1 to 18 and presses the random number button on her calculator to obtain a 3-digit decimal. Millie then multiplies the first digit in this decimal by two and chooses the person corresponding to this new number. Decimals in which the first digit is zero are ignored.

1. Give a reason why this is not a satisfactory method of choosing a person. Millie obtained a random sample of 5 people of her own age by a satisfactory sampling method and found that their heights in metres were \(1.66,1.68,1.54,1.65\) and 1.57 . Heights are known to be normally distributed with variance \(0.0052 \mathrm {~m} ^ { 2 }\).
2. Find a \(98 \%\) confidence interval for the mean height of people of Millie's age.

2 The editor of a magazine wishes to obtain the views of a random sample of readers about the future of the magazine.

1. A sub-editor proposes that they include in one issue of the magazine a questionnaire for readers to complete and return. Give two reasons why the readers who return the questionnaire would not form a random sample. The editor decides to use a table of random numbers to select a random sample of 50 readers from the 7302 regular readers. These regular readers are numbered from 1 to 7302 . The first few random numbers which the editor obtains from the table are as follows. $$49757 \quad 80239 \quad 52038 \quad 60882$$
2. Use these random numbers to select the first three members in the sample.

6 In order to obtain a random sample of people who live in her town, Jane chooses people at random from the telephone directory for her town.

1. Give a reason why Jane's method will not give a random sample of people who live in the town. Jane now uses a valid method to choose a random sample of 200 people from her town and finds that 38 live in apartments.
2. Calculate an approximate \(99 \%\) confidence interval for the proportion of all people in Jane's town who live in apartments.
3. Jane uses the same sample to give a confidence interval of width 0.1 for this proportion. This interval is an \(x \%\) confidence interval. Find the value of \(x\).

4 In a survey a random sample of 150 households in Nantville were asked to fill in a questionnaire about household budgeting.

1. The results showed that 33 households owned more than one car. Find an approximate \(99 \%\) confidence interval for the proportion of all households in Nantville with more than one car. [4]
2. The results also included the weekly expenditure on food, \(x\) dollars, of the households. These were summarised as follows. $$n = 150 \quad \Sigma x = 19035 \quad \Sigma x ^ { 2 } = 4054716$$ Find unbiased estimates of the mean and variance of the weekly expenditure on food of all households in Nantville.
3. The government has a list of all the households in Nantville numbered from 1 to 9526. Describe briefly how to use random numbers to select a sample of 150 households from this list.

2 Dominic wishes to choose a random sample of five students from the 150 students in his year. He numbers the students from 1 to 150 . Then he uses his calculator to generate five random numbers between 0 and 1 . He multiplies each random number by 150 and rounds up to the next whole number to give a student number.

1. Dominic's first random number is 0.392 . Find the student number that is produced by this random number.
2. Dominic's second student number is 104 . Find a possible random number that would produce this student number.
3. Explain briefly why five random numbers may not be enough to produce a sample of five student numbers.

2 A school has 900 pupils. For a survey, Jan obtains a list of all the pupils, numbered 1 to 900 in alphabetical order. She then selects a sample by the following method. Two fair dice, one red and one green, are thrown, and the number in the list of the first pupil in the sample is determined by the following table. For example, if the scores on the red and green dice are 5 and 2 respectively, then the first member of the sample is the pupil numbered 8 in the list. Starting with this first number, every 12th number on the list is then used, so that if the first pupil selected is numbered 8 , the others will be numbered \(20,32,44 , \ldots\).

1. State the size of the sample.
2. Explain briefly whether the following statements are true.
   1. Each pupil in the school has an equal probability of being in the sample.
   2. The pupils in the sample are selected independently of one another.
   3. Give a reason why the number of the first pupil in the sample should not be obtained simply by adding together the scores on the two dice. Justify your answer.

2 A village has a population of 600 people. A sample of 12 people is obtained as follows. A list of all 600 people is obtained and a three-digit number, between 001 and 600 inclusive, is allocated to each name in alphabetical order. Twelve three-digit random numbers, between 001 and 600 inclusive, are obtained and the people whose names correspond to those numbers are chosen.

1. Find the probability that all 12 of the numbers chosen are 500 or less.
2. When the selection has been made, it is found that all of the numbers chosen are 500 or less. One of the people in the village says, "The sampling method must have been biased." Comment on this statement.

1 It is desired to obtain a random sample of 15 pupils from a large school. One pupil suggests listing all the pupils in the school in alphabetical order and choosing the first 15 names on the list.

1. Explain why this method is unsatisfactory.
2. Suggest a better method.

2 A certain neighbourhood contains many small houses (with small gardens) and a few large houses (with large gardens). A sample survey of all houses is to be carried out in this neighbourhood. A student suggests that the sample could be selected by sticking a pin into a map of the neighbourhood the requisite number of times, while blindfolded.

1. Give two reasons why this method does not produce a random sample.
2. Describe a better method.