2.01a Population and sample: terminology

6 Employees at a particular company have been working seven hours each day, from 9 am to 4 pm . To try to reduce absence, the company decides to introduce 'flexi-time' and allow employees to work their seven hours each day at any time between 7 am and 9 pm . For a random sample of 10 employees, the numbers of hours of absence in the year before and the year after the introduction of flexi-time are given in the following table. Use a paired sample \(t\)-test to test, at the \(10 \%\) significance level, whether the population mean number of hours of absence has decreased, following the introduction of flexi-time.

5 The mean breaking strength of cables made at a certain factory is supposed to be 5 tonnes. The quality control department wishes to test whether the mean breaking strength of cables made by a particular machine is actually less than it should be. They take a random sample of 60 cables. For each cable they find the breaking strength by gradually increasing the tension in the cable and noting the tension when the cable breaks.

1. Give a reason why it is necessary to take a sample rather then testing all the cables produced by the machine.
2. The mean breaking strength of the 60 cables in the sample is found to be 4.95 tonnes. Given that the population standard deviation of breaking strengths is 0.15 tonnes, test at the \(1 \%\) significance level whether the population mean breaking strength is less than it should be.
3. Explain whether it was necessary to use the Central Limit theorem in the solution to part (ii).

2 A researcher is investigating the lengths, in kilometres, of the journeys to work of the employees at a certain firm. She takes a random sample of 10 employees.

1. State what is meant by 'random' in this context. The results of her sample are as follows. $$\begin{array} { l l l l l l l l l l } 1.5 & 2.0 & 3.6 & 5.9 & 4.8 & 8.7 & 3.5 & 2.9 & 4.1 & 3.0 \end{array}$$
2. Find unbiased estimates of the population mean and variance.
3. State what is meant by 'population' in this context.

3

1. Give a reason for using a sample rather than the whole population in carrying out a statistical investigation.
2. Tennis balls of a certain brand are known to have a mean height of bounce of 64.7 cm , when dropped from a height of 100 cm . A change is made in the manufacturing process and it is required to test whether this change has affected the mean height of bounce. 100 new tennis balls are tested and it is found that their mean height of bounce when dropped from a height of 100 cm is 65.7 cm and the unbiased estimate of the population variance is \(15 \mathrm {~cm} ^ { 2 }\).
   1. Calculate a \(95 \%\) confidence interval for the population mean.
   2. Use your answer to part (ii) (a) to explain what conclusion can be drawn about whether the change has affected the mean height of bounce.

3 Each of a random sample of 15 students was asked how long they spent revising for an exam. The results, in minutes, were as follows. $$\begin{array} { l l l l l l l l l l l l l l l } 50 & 70 & 80 & 60 & 65 & 110 & 10 & 70 & 75 & 60 & 65 & 45 & 50 & 70 & 50 \end{array}$$ Assume that the times for all students are normally distributed with mean \(\mu\) minutes and standard deviation 12 minutes.

1. Calculate a \(92 \%\) confidence interval for \(\mu\).
2. Explain what is meant by a \(92 \%\) confidence interval for \(\mu\).
3. Explain what is meant by saying that a sample is 'random'.

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.

2 In a survey, a random sample of 250 adults in Fromleigh were asked to fill in a questionnaire about their travel.

1. It was found that 102 adults in the sample travel by bus. Find an approximate \(90 \%\) confidence interval for the proportion of all the adults in Fromleigh who travel by bus.
2. The survey included a question about the amount, \(x\) dollars, spent on travel per year. The results are summarised as follows. $$n = 250 \quad \Sigma x = 50460 \quad \Sigma x ^ { 2 } = 19854200$$ Find unbiased estimates of the population mean and variance of the amount spent per year on travel.
   A councillor wanted to select a random sample of houses in Fromleigh. He planned to select the first house on each of the 143 streets in Fromleigh.
3. Explain why this would not provide a random sample.

4

1. Give a reason why, in carrying out a statistical investigation, a sample rather than a complete population may be used.
2. Rose wishes to investigate whether men in her town have a different life-span from the national average of 71.2 years. She looks at government records for her town and takes a random sample of the ages of 110 men who have died recently. Their mean age in years was 69.3 and the unbiased estimate of the population variance was 65.61.
   1. Calculate a \(90 \%\) confidence interval for the population mean and explain what you understand by this confidence interval.
   2. State with a reason what conclusion about the life-span of men in her town Rose could draw from this confidence interval.

3

1. Explain what is meant by the term 'random sample'. In a random sample of 350 food shops it was found that 130 of them had Special Offers.
2. Calculate an approximate \(95 \%\) confidence interval for the proportion of all food shops with Special Offers.
3. Estimate the size of a random sample required for an approximate \(95 \%\) confidence interval for this proportion to have a width of 0.04 .

7

1. Give a reason why sampling would be required in order to reach a conclusion about
   1. the mean height of adult males in England,
   2. the mean weight that can be supported by a single cable of a certain type without the cable breaking.
2. The weights, in kg , of sacks of potatoes are represented by the random variable \(X\) with mean \(\mu\) and standard deviation \(\sigma\). The weights of a random sample of 500 sacks of potatoes are found and the results are summarised below. $$n = 500 , \quad \Sigma x = 9850 , \quad \Sigma x ^ { 2 } = 194125 .$$
   1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
   2. A further random sample of 60 sacks of potatoes is taken. Using your values from part (b) (i), find the probability that the mean weight of this sample exceeds 19.73 kg .
   3. Explain whether it was necessary to use the Central Limit Theorem in your calculation in part (b) (ii).

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.

4

1. Explain briefly what is meant by a random sample. Random numbers are used to select, with replacement, a sample of size \(n\) from a population numbered 000, 001, 002, ..., 799.
2. If \(n = 6\), find the probability that exactly 4 of the selected sample have numbers less than 500 .
3. If \(n = 60\), use a suitable approximation to calculate the probability that at least 40 of the selected sample have numbers less than 500 .

2. A rugby club coach uses club records to take a random sample of 15 players from 1990 and an independent random sample of 15 players from 2010. The body weight of each player was recorded to the nearest kg and the results from 2010 are summarised in the table below.

1. Find the estimated values in kg of the summary statistics \(a\), \(b\) and \(c\) in the table below.

   	Estimate in 1990	Estimate in 2010
   Mean	83.0	\(a\)
   Median	82.0	\(b\)
   Variance	44.0	\(c\)

   Give your answers to 3 significant figures. The rugby coach claims that players' body weight increased between 1990 and 2010.
2. Using the table in part (a), comment on the rugby coach's claim. \includegraphics[max width=\textwidth, alt={}, center]{a839a89a-17f0-473b-ac10-bcec3dbe97f7-05_104_97_2613_1784}

1

1. Define simple random sampling. Describe briefly one difficulty associated with simple random sampling.
2. Freeze-drying is an economically important process used in the production of coffee. It improves the retention of the volatile aroma compounds. In order to maintain the quality of the coffee, technologists need to monitor the drying rate, measured in suitable units, at regular intervals. It is known that, for best results, the mean drying rate should be 70.3 units and anything substantially less than this would be detrimental to the coffee. Recently, a random sample of 12 observations of the drying rate was as follows. $$\begin{array} { l l l l l l l l l l l l } 66.0 & 66.1 & 59.8 & 64.0 & 70.9 & 71.4 & 66.9 & 76.2 & 65.2 & 67.9 & 69.2 & 68.5 \end{array}$$
   1. Carry out a test to investigate at the \(5 \%\) level of significance whether the mean drying rate appears to be less than 70.3. State the distributional assumption that is required for this test.
   2. Find a 95\% confidence interval for the true mean drying rate.

4

1. At a college, two examiners are responsible for marking, independently, the students' projects. Each examiner awards a mark out of 100 to each project. There is some concern that the examiners' marks do not agree, on average. Consequently a random sample of 12 projects is selected and the marks awarded to them are compared.
   1. Describe how a random sample of projects should be chosen.
   2. The marks given for the projects in the sample are as follows.

      Project	1	2	3	4	5	6	7	8	9	10	11	12
      Examiner A	58	37	72	78	67	77	62	41	80	60	65	70
      Examiner B	73	47	74	71	78	96	54	27	97	73	60	66

      Carry out a test at the \(10 \%\) level of significance of the hypotheses \(\mathrm { H } _ { 0 } : m = 0 , \mathrm { H } _ { 1 } : m \neq 0\), where \(m\) is the population median difference.
2. A calculator has a built-in random number function which can be used to generate a list of random digits. If it functions correctly then each digit is equally likely to be generated. When it was used to generate 100 random digits, the frequencies of the digits were as follows.

   Digit	0	1	2	3	4	5	6	7	8	9
   Frequency	6	8	11	14	12	9	15	5	14	6

   Use a goodness of fit test, with a significance level of \(10 \%\), to investigate whether the random number function is generating digits with equal probability.

2

1. 1. Give two reasons why an investigator might need to take a sample in order to obtain information about a population.
   2. State two requirements of a sample.
   3. Discuss briefly the advantage of the sampling being random.
2. 1. Under what circumstances might one use a Wilcoxon single sample test in order to test a hypothesis about the median of a population? What distributional assumption is needed for the test?
   2. On a stretch of road leading out of the centre of a town, highways officials have been monitoring the speed of the traffic in case it has increased. Previously it was known that the median speed on this stretch was 28.7 miles per hour. For a random sample of 12 vehicles on the stretch, the following speeds were recorded. $$\begin{array} { l l l l l l l l l l l l } 32.0 & 29.1 & 26.1 & 35.2 & 34.4 & 28.6 & 32.3 & 28.5 & 27.0 & 33.3 & 28.2 & 31.9 \end{array}$$ Carry out a test, with a \(5 \%\) significance level, to see whether the speed of the traffic on this stretch of road seems to have increased on the whole.
      [0pt] [10]

2

1. Explain what is meant by a simple random sample. A manufacturer produces tins of paint which nominally contain 1 litre. The quantity of paint delivered by the machine that fills the tins can be assumed to be a Normally distributed random variable. The machine is designed to deliver an average of 1.05 litres to each tin. However, over time paint builds up in the delivery nozzle of the machine, reducing the quantity of paint delivered. Random samples of 10 tins are taken regularly from the production process. If a significance test, carried out at the \(5 \%\) level, suggests that the average quantity of paint delivered is less than 1.02 litres, the machine is cleaned.
2. By carrying out an appropriate test, determine whether or not the sample below leads to the machine being cleaned. $$\begin{array} { l l l l l l l l l l } 0.994 & 1.010 & 1.021 & 1.015 & 1.016 & 1.022 & 1.009 & 1.007 & 1.011 & 1.026 \end{array}$$ Each time the machine has been cleaned, a random sample of 10 tins is taken to determine whether or not the average quantity of paint delivered has returned to 1.05 litres.
3. On one occasion after the machine has been cleaned, the quality control manager thinks that the distribution of the quantity of paint is symmetrical but not necessarily Normal. The sample on this occasion is as follows.

   1.055	1.064	1.063	1.043	1.062	1.070	1.059	1.044	1.054
   1.053

   By carrying out an appropriate test at the \(5 \%\) level of significance, determine whether or not this sample supports the conclusion that the average quantity of paint delivered is 1.05 litres.

1. Jerry is studying visibility for Camborne using the large data set June 1987.

The table below contains two extracts from the large data set.
It shows the daily maximum relative humidity and the daily mean visibility. (The units for Daily Mean Visibility are deliberately omitted.)
Given that daily mean visibility is given to the nearest 100,

1. write down the range of distances in metres that corresponds to the recorded value 0 for the daily mean visibility. Jerry drew the following scatter diagram, Figure 2, and calculated some statistics using the June 1987 data for Camborne from the large data set. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d62e5a00-cd23-417f-b244-8b3e24da4aa2-04_823_1764_1281_137} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Jerry defines an outlier as a value that is more than 1.5 times the interquartile range above \(Q _ { 3 }\) or more than 1.5 times the interquartile range below \(Q _ { 1 }\).
2. Show that the point circled on the scatter diagram is an outlier for visibility.
3. Interpret the correlation between the daily mean visibility and the daily maximum relative humidity. Jerry drew the following scatter diagram, Figure 3, using the June 1987 data for Camborne from the large data set, but forgot to label the \(x\)-axis.
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d62e5a00-cd23-417f-b244-8b3e24da4aa2-05_730_1056_342_386} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure}
4. Using your knowledge of the large data set, suggest which variable the \(x\)-axis on this scatter diagram represents.

1. Jiang is studying the variable Daily Mean Pressure from the large data set.

He drew the following box and whisker plot for these data for one of the months for one location using a linear scale but

• he failed to label all the values on the scale
• he gave an incorrect value for the median \includegraphics[max width=\textwidth, alt={}, center]{08e3b0b0-2155-4b37-83e3-343c317ca10c-09_248_1264_573_402}

Daily Mean Pressure (hPa)
Using your knowledge of the large data set, suggest a suitable value for

1. the median,
2. the range.
   (You are not expected to have memorised values from the large data set. The question is simply looking for sensible answers.)
   1. Jiang is studying the variable Daily Mean Pressure from the large data set.
   He drew the following box and whisker plot for these data for one of the months for one nong asing at
   • he gave an incorrect value for the median "
   Using your knowledge of the large data set, suggest a suitable value for
3. the median,
   " \includegraphics[max width=\textwidth, alt={}, center]{08e3b0b0-2155-4b37-83e3-343c317ca10c-09_42_31_1213_1304}
   "

1. Fred and Nadine are investigating whether there is a linear relationship between Daily Mean Pressure, \(p \mathrm { hPa }\), and Daily Mean Air Temperature, \(t ^ { \circ } \mathrm { C }\), in Beijing using the 2015 data from the large data set.

Fred randomly selects one month from the data set and draws the scatter diagram in Figure 1 using the data from that month. The scale has been left off the horizontal axis. \begin{figure}[h] \end{figure}

1. Describe the correlation shown in Figure 1. Nadine chooses to use all of the data for Beijing from 2015 and draws the scatter diagram in Figure 2. She uses the same scales as Fred. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{854568d2-b32d-44de-8a9c-26372e509c20-04_777_1509_1841_278} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure}
2. Explain, in context, what Nadine can infer about the relationship between \(p\) and \(t\) using the information shown in Figure 2.
3. Using your knowledge of the large data set, state a value of \(p\) for which interpolation can be used with Figure 2 to predict a value of \(t\).
4. Using your knowledge of the large data set, explain why it is not meaningful to look for a linear relationship between Daily Mean Wind Speed (Beaufort Conversion) and Daily Mean Air Temperature in Beijing in 2015.
5. Explain, in context, what Nadine can infer about the relationship between \(p\) and \(t\) using the information shown in Figure 2.

2. \begin{figure}[h] \end{figure} The partially completed box plot in Figure 1 shows the distribution of daily mean air temperatures using the data from the large data set for Beijing in 2015 An outlier is defined as a value
more than \(1.5 \times\) IQR below \(Q _ { 1 }\) or
more than \(1.5 \times\) IQR above \(Q _ { 3 }\) The three lowest air temperatures in the data set are \(7.6 ^ { \circ } \mathrm { C } , 8.1 ^ { \circ } \mathrm { C }\) and \(9.1 ^ { \circ } \mathrm { C }\) The highest air temperature in the data set is \(32.5 ^ { \circ } \mathrm { C }\)

1. Complete the box plot in Figure 1 showing clearly any outliers.
2. Using your knowledge of the large data set, suggest from which month the two outliers are likely to have come. Using the data from the large data set, Simon produced the following summary statistics for the daily mean air temperature, \(x ^ { \circ } \mathrm { C }\), for Beijing in 2015 $$n = 184 \quad \sum x = 4153.6 \quad \mathrm {~S} _ { x x } = 4952.906$$
3. Show that, to 3 significant figures, the standard deviation is \(5.19 ^ { \circ } \mathrm { C }\) Simon decides to model the air temperatures with the random variable $$T \sim \mathrm {~N} \left( 22.6,5.19 ^ { 2 } \right)$$
4. Using Simon's model, calculate the 10th to 90th interpercentile range. Simon wants to model another variable from the large data set for Beijing using a normal distribution.
5. State two variables from the large data set for Beijing that are not suitable to be modelled by a normal distribution. Give a reason for each answer. \includegraphics[max width=\textwidth, alt={}, center]{d1eaaae7-c1dc-4aee-ab54-59f35519a7a4-09_473_1813_2161_127}
   (Total for Question 2 is 11 marks)

10 Jane conducted a survey. She chose a sample of people from three towns, A, B and C. She noted the following information. 400 people were chosen.
230 people were adults.
55 adults were from town A .
65 children were from town A .
35 children were from town B .
150 people were from town B .

1. In the Printed Answer Booklet, complete the two-way frequency table.

   \multirow{2}{*}{}	Town
   	A	B	C	Total
   adult
   child
   Total

2. One of the people is chosen at random.
   1. Find the probability that this person is an adult from town A .
   2. Given that the person is from town A , find the probability that the person is an adult. For another survey, Jane wanted to choose a random sample from the 820 students living in a particular hostel. She numbered the students from 1 to 820 and then generated some random numbers on her calculator. The random numbers were 0.114287562 and 0.081859817 . Jane's friend Kareem used these figures to write down the following sample of five student numbers. 114, 142, 428, 287 and 756 Jane used the same figures to write down the following sample of five student numbers.
      114, 287, 562, 81 and 817
3. 1. State, with a reason, which one of these samples is not random.
   2. Explain why Jane omitted the number 859 from her sample.

6 An app on my new smartphone records the number of times in a day I use the phone. The data for each day since I bought the phone are shown in the stem and leaf diagram. Key: 3|1 means 31

1. Explain whether these data are a sample or a population.
2. Describe the shape of the distribution.
3. Determine the interquartile range.
4. Use your answer to part (c) to determine whether there are any outliers in the lower tail.

15 Ali and Sam are playing a game in which Ali tosses a coin 5 times. If there are 4 or 5 heads, Ali wins the game. Otherwise Sam wins the game. They decide to play the game 50 times.

1. Initially Sam models the situation by assuming the coin is fair. Determine the number of games Ali is expected to win according to this model. Ali thinks the coin may be biased, with probability \(p\) of obtaining heads when the coin is tossed. Before playing the game, Ali and Sam decide to collect some data to estimate the value of \(p\). Sam tosses the coin 15 times and records the number of heads obtained. Ali tosses the coin 25 times and records the number of heads obtained.
2. Explain why it is better to use the combined data rather than just Sam's data or just Ali's data to estimate the value of \(p\). Ali records 20 heads and Sam records 8 heads.
3. Use the combined data to estimate the value of \(p\). Ali now models the situation using the value of \(p\) found in part (c) as the probability of obtaining heads when the coin is tossed.
4. Determine how many games Ali expects to win using this value of \(p\) to model the situation.
5. Ali wins 25 of the 50 games. Explain whether Sam's model or Ali's model is a better fit for the data. \section*{END OF QUESTION PAPER} }{www.ocr.org.uk}) after the live examination series.
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