Expected frequency with unknown parameter

4 A mathematical puzzle is given to a large number of students. The times taken to complete the puzzle are normally distributed with mean 14.6 minutes and standard deviation 5.2 minutes.

1. In a random sample of 250 of the students, how many would you expect to have taken more than 20 minutes to complete the puzzle?
   All the students are given a second puzzle to complete. Their times, in minutes, are normally distributed with mean \(\mu\) and standard deviation \(\sigma\). It is found that \(20 \%\) of the students have times less than 14.5 minutes and \(67 \%\) of the students have times greater than 18.5 minutes.
2. Find the value of \(\mu\) and the value of \(\sigma\).

5 The lengths of Western bluebirds are normally distributed with mean 16.5 cm and standard deviation 0.6 cm . A random sample of 150 of these birds is selected.

1. How many of these 150 birds would you expect to have length between 15.4 cm and 16.8 cm ?
   The lengths of Eastern bluebirds are normally distributed with mean 18.4 cm and standard deviation \(\sigma \mathrm { cm }\). It is known that \(72 \%\) of Eastern bluebirds have length greater than 17.1 cm .
2. Find the value of \(\sigma\).
   A random sample of 120 Eastern bluebirds is chosen.
3. Use an approximation to find the probability that fewer than 80 of these 120 bluebirds have length greater than 17.1 cm .

4 A company sells small and large bags of rice. The masses of the small bags of rice are normally distributed with mean 1.20 kg and standard deviation 0.16 kg .

1. In a random sample of 500 of these small bags of rice, how many would you expect to have a mass greater than 1.26 kg ?
   The masses of the large bags of rice are normally distributed with mean 2.50 kg and standard deviation \(\sigma \mathrm { kg } .20 \%\) of these large bags of rice have a mass less than 2.40 kg .
2. Find the value of \(\sigma\).
   A random sample of 80 large bags of rice is chosen.
3. Use a suitable approximation to find the probability that fewer than 22 of these large bags of rice have a mass less than 2.40 kg .

4 The heights, in metres, of white pine trees are normally distributed with mean 19.8 and standard deviation 2.4 . In a certain forest there are 450 white pine trees.

1. How many of these trees would you expect to have height less than 18.2 metres?
   The heights, in metres, of red pine trees are normally distributed with mean 23.4 and standard deviation \(\sigma\). It is known that \(26 \%\) of red pine trees have height greater than 25.5 metres.
2. Find the value of \(\sigma\).

7

1. Once a week Zak goes for a run. The time he takes, in minutes, has a normal distribution with mean 35.2 and standard deviation 4.7.
   1. Find the expected number of days during a year ( 52 weeks) for which Zak takes less than 30 minutes for his run.
   2. The probability that Zak's time is between 35.2 minutes and \(t\) minutes, where \(t > 35.2\), is 0.148 . Find the value of \(t\).
2. The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). It is given that \(\mathrm { P } ( X < 7 ) = 0.2119\) and \(\mathrm { P } ( X < 10 ) = 0.6700\). Find the values of \(\mu\) and \(\sigma\).

5 The heights of school desks have a normal distribution with mean 69 cm and standard deviation \(\sigma \mathrm { cm }\). It is known that 15.5\% of these desks have a height greater than 70 cm .

1. Find the value of \(\sigma\). When Jodu sits at a desk, his knees are at a height of 58 cm above the floor. A desk is comfortable for Jodu if his knees are at least 9 cm below the top of the desk. Jodu's school has 300 desks.
2. Calculate an estimate of the number of these desks that are comfortable for Jodu.

3

1. The volume of soup in Super Soup cartons has a normal distribution with mean \(\mu\) millilitres and standard deviation 9 millilitres. Tests have shown that \(10 \%\) of cartons contain less than 440 millilitres of soup. Find the value of \(\mu\).
2. A food retailer orders 150 Super Soup cartons. Calculate the number of these cartons for which you would expect the volume of soup to be more than 1.8 standard deviations above the mean.

5

1. Give an example of a variable in real life which could be modelled by a normal distribution.
2. The random variable \(X\) is normally distributed with mean \(\mu\) and variance 21.0. Given that \(\mathrm { P } ( X > 10.0 ) = 0.7389\), find the value of \(\mu\).
3. If 300 observations are taken at random from the distribution in part (ii), estimate how many of these would be greater than 22.0.

7

1. A petrol station finds that its daily sales, in litres, are normally distributed with mean 4520 and standard deviation 560.
   1. Find on how many days of the year ( 365 days) the daily sales can be expected to exceed 3900 litres. The daily sales at another petrol station are \(X\) litres, where \(X\) is normally distributed with mean \(m\) and standard deviation 560. It is given that \(\mathrm { P } ( X > 8000 ) = 0.122\).
   2. Find the value of \(m\).
   3. Find the probability that daily sales at this petrol station exceed 8000 litres on fewer than 2 of 6 randomly chosen days.
2. The random variable \(Y\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\). Given that \(\sigma = \frac { 2 } { 3 } \mu\), find the probability that a random value of \(Y\) is less than \(2 \mu\).

4 The time taken for cucumber seeds to germinate under certain conditions has a normal distribution with mean 125 hours and standard deviation \(\sigma\) hours.

1. It is found that \(13 \%\) of seeds take longer than 136 hours to germinate. Find the value of \(\sigma\).
2. 170 seeds are sown. Find the expected number of seeds which take between 131 and 141 hours to germinate.

6. The waiting time, \(L\) minutes, to see a doctor at a health centre is normally distributed with \(L \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\). Given that \(\mathrm { P } ( L < 15 ) = 0.9\) and \(\mathrm { P } ( L < 5 ) = 0.05\)

1. find the value of \(\mu\) and the value of \(\sigma\). There are 23 people waiting to see a doctor at the health centre.
2. Determine the expected number of these people who will have a waiting time of more than 12 minutes.

8. The lifetimes of bulbs used in a lamp are normally distributed. A company \(X\) sells bulbs with a mean lifetime of 850 hours and a standard deviation of 50 hours.

1. Find the probability of a bulb, from company \(X\), having a lifetime of less than 830 hours.
2. In a box of 500 bulbs, from company \(X\), find the expected number having a lifetime of less than 830 hours. A rival company \(Y\) sells bulbs with a mean lifetime of 860 hours and \(20 \%\) of these bulbs have a lifetime of less than 818 hours.
3. Find the standard deviation of the lifetimes of bulbs from company \(Y\). Both companies sell the bulbs for the same price.
4. State which company you would recommend. Give reasons for your answer.

1. The weight, in grams, of beans in a tin is normally distributed with mean \(\mu\) and standard deviation 7.8

Given that \(10 \%\) of tins contain less than 200 g , find

1. the value of \(\mu\)
2. the percentage of tins that contain more than 225 g of beans. The machine settings are adjusted so that the weight, in grams, of beans in a tin is normally distributed with mean 205 and standard deviation \(\sigma\).
3. Given that \(98 \%\) of tins contain between 200 g and 210 g find the value of \(\sigma\).

8. The lifetimes of bulbs used in a lamp are normally distributed. A company \(X\) sells bulbs with a mean lifetime of 850 hours and a standard deviation of 50 hours.

1. Find the probability of a bulb, from company \(X\), having a lifetime of less than 830 hours.
2. In a box of 500 bulbs, from company \(X\), find the expected number having a lifetime of less than 830 hours. A rival company \(Y\) sells bulbs with a mean lifetime of 860 hours and \(20 \%\) of these bulbs have a lifetime of less than 818 hours.
3. Find the standard deviation of the lifetimes of bulbs from company \(Y\). Both companies sell the bulbs for the same price.
4. State which company you would recommend. Give reasons for your answer. Mathematical Formulae (Pink or Green) Nil \section*{Wednesday 13 January 2010 - Afternoon} Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them. In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Statistics S1), the paper reference (6683), your surname, other name and signature.
   Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
   Full marks may be obtained for answers to ALL questions.
   The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2). There are 7 questions on this paper. The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled.
   You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
   1. A jar contains 2 red, 1 blue and 1 green bead. Two beads are drawn at random from the jar without replacement.
   2. Draw a tree diagram to illustrate all the possible outcomes and associated probabilities. State your probabilities clearly.
   3. Find the probability that a blue bead and a green bead are drawn from the jar.
   4. The 19 employees of a company take an aptitude test. The scores out of 40 are illustrated in the stem and leaf diagram below.
   where \(a\) is a constant.
5. Find the value of \(a\).
6. Write down \(\mathrm { E } ( X )\).
7. Find \(\operatorname { Var } ( X )\). The random variable \(Y = 6 - 2 X\).
8. Find \(\operatorname { Var } ( Y )\).
9. Calculate \(\mathrm { P } ( X \geq Y )\). Shivani selects a ball and spins the appropriate coin.
10. Find the probability that she obtains a head. Given that Tom selected a ball at random and obtained a head when he spun the appropriate coin,
11. find the probability that Tom selected a red ball. Shivani and Tom each repeat this experiment.
12. Find the probability that the colour of the ball Shivani selects is the same as the colour of the ball Tom selects.
    4. The Venn diagram in Figure 1 shows the number of students in a class who read any of 3 popular magazines \(A , B\) and \(C\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e8afd947-55ac-424b-8db5-d5aa856ef4d7-064_264_615_360_443} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} One of these students is selected at random.
13. Show that the probability that the student reads more than one magazine is \(\frac { 1 } { 6 }\).
14. Find the probability that the student reads \(A\) or \(B\) (or both).
15. Write down the probability that the student reads both \(A\) and \(C\). Given that the student reads at least one of the magazines,
16. find the probability that the student reads \(C\).
17. Determine whether or not reading magazine \(B\) and reading magazine \(C\) are statistically independent.
    5. A teacher selects a random sample of 56 students and records, to the nearest hour, the time spent watching television in a particular week. where \(a , b\) and \(c\) are constants.
    The cumulative distribution function \(\mathrm { F } ( y )\) of \(Y\) is given in the following table.
18. Estimate the number of motorists who were delayed between 8.5 and 13.5 minutes by the roadworks.
    (2)
    2. (a) State in words the relationship between two events \(R\) and \(S\) when \(\mathrm { P } ( R \cap S ) = 0\). The events \(A\) and \(B\) are independent with \(\mathrm { P } ( A ) = \frac { 1 } { 4 }\) and \(\mathrm { P } ( A \cup B ) = \frac { 2 } { 3 }\).
    Find
19. \(\mathrm { P } ( B )\),
20. \(\mathrm { P } \left( A ^ { \prime } \cap B \right)\),
21. \(\mathrm { P } \left( B ^ { \prime } \mid A \right)\).
    3. The discrete random variable \(X\) can take only the values \(2,3,4\) or 6 . For these values the probability distribution function is given by [You may use \(\sum p ^ { 2 } = 1967\) and \(\sum p t = 694\) ]
22. On graph paper, draw a scatter diagram to represent these data.
23. Explain why a linear regression model may be appropriate to describe the relationship between \(p\) and \(t\).
24. Calculate the value of \(S _ { p t }\) and the value of \(S _ { p p }\).
25. Find the equation of the regression line of \(t\) on \(p\), giving your answer in the form \(t = a + b p\).
26. Plot the point \(( \bar { p } , \bar { t } )\) and draw the regression line on your scatter diagram. The scientist reviews similar studies and finds that pollutant levels above 16 are likely to result in the death of a chick soon after hatching.
27. Estimate the minimum thinning of the shell that is likely to result in the death of a chick.
    4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e8afd947-55ac-424b-8db5-d5aa856ef4d7-077_401_741_296_1761} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows how 25 people travelled to work.
    Their travel to work is represented by the events
    B bicycle
    \(T \quad\) train
    \(W\) walk
28. Write down 2 of these events that are mutually exclusive. Give a reason for your answer.
29. Determine whether or not \(B\) and \(T\) are independent events. One person is chosen at random.
    Find the probability that this person
30. walks to work,
31. travels to work by bicycle and train. Given that this person travels to work by bicycle,
32. find the probability that they will also take the train.
    5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e8afd947-55ac-424b-8db5-d5aa856ef4d7-078_618_812_301_315} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} A policeman records the speed of the traffic on a busy road with a 30 mph speed limit.
    He records the speeds of a sample of 450 cars. The histogram in Figure 2 represents the results.
33. Calculate the number of cars that were exceeding the speed limit by at least 5 mph in the sample.
34. Estimate the value of the mean speed of the cars in the sample.
35. Estimate, to 1 decimal place, the value of the median speed of the cars in the sample.
36. Comment on the shape of the distribution. Give a reason for your answer.
37. State, with a reason, whether the estimate of the mean or the median is a better representation of the average speed of the traffic on the road.
    6. The heights of an adult female population are normally distributed with mean 162 cm and standard deviation 7.5 cm .
38. Find the probability that a randomly chosen adult female is taller than 150 cm .
    (3) Sarah is a young girl. She visits her doctor and is told that she is at the 60th percentile for height.
39. Assuming that Sarah remains at the 60th percentile, estimate her height as an adult.
    (3) The heights of an adult male population are normally distributed with standard deviation 9.0 cm .
    Given that \(90 \%\) of adult males are taller than the mean height of adult females,
40. find the mean height of an adult male.
    (4)
    7. A manufacturer carried out a survey of the defects in their soft toys. It is found that the probability of a toy having poor stitching is 0.03 and that a toy with poor stitching has a probability of 0.7 of splitting open. A toy without poor stitching has a probability of 0.02 of splitting open.
41. Draw a tree diagram to represent this information.
    (3)
42. Find the probability that a randomly chosen soft toy has exactly one of the two defects, poor stitching or splitting open.
    (3) The manufacturer also finds that soft toys can become faded with probability 0.05 and that this defect is independent of poor stitching or splitting open. A soft toy is chosen at random.
43. Find the probability that the soft toy has none of these 3 defects.
44. Find the probability that the soft toy has exactly one of these 3 defects. \section*{Advanced Level} \section*{Friday 18 January 2013 - Afternoon} Nil Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulas stored in them. In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Statistics S1), the paper reference (6683), your surname, other name and signature.
    Values from the statistical tables should be quoted in full. When a calculator is used, the answer should be given to an appropriate degree of accuracy. A booklet 'Mathematical Formulae and Statistical Tables' is provided.
    Full marks may be obtained for answers to ALL questions.
    This paper has 7 questions.
    The total mark for this paper is 75 . You must ensure that your answers to parts of questions are clearly labelled.
    You must show sufficient working to make your methods clear to the Examiner.
    Answers without working may not gain full credit. \section*{P41805A}
    1. A teacher asked a random sample of 10 students to record the number of hours of television, \(t\), they watched in the week before their mock exam. She then calculated their grade, \(g\), in their mock exam. The results are summarised as follows.
    $$\sum t = 258 \quad \sum t ^ { 2 } = 8702 \quad \sum g = 63.6 \quad \mathrm {~S} _ { g g } = 7.864 \quad \sum g t = 1550.2$$
45. Find \(\mathrm { S } _ { t t }\) and \(\mathrm { S } _ { g t }\).
46. Calculate, to 3 significant figures, the product moment correlation coefficient between \(t\) and \(g\). The teacher also recorded the number of hours of revision, \(v\), these 10 students completed during the week before their mock exam. The correlation coefficient between \(t\) and \(v\) was - 0.753 .
47. Describe, giving a reason, the nature of the correlation you would expect to find between \(v\) and \(g\).
    2. The discrete random variable \(X\) can take only the values 1,2 and 3 . For these values the cumulative distribution function is defined by $$\mathrm { F } ( x ) = \frac { x ^ { 3 } + k } { 40 } , \quad x = 1,2,3 .$$
48. Show that \(k = 13\).
49. Find the probability distribution of \(X\). Given that \(\operatorname { Var } ( X ) = \frac { 259 } { 320 }\),
50. find the exact value of \(\operatorname { Var } ( 4 X - 5 )\).
    3. A biologist is comparing the intervals ( \(m\) seconds) between the mating calls of a certain species of tree frog and the surrounding temperature ( \(t ^ { \circ } \mathrm { C }\) ). The following results were obtained. [You may assume that \(\sum h = 7150 , \sum t = 110 , \sum h ^ { 2 } = 7171500 , \sum t ^ { 2 } = 1716 , \sum t h = 64980\) and \(\mathrm { S } t t = 371.56\) ]
51. Calculate \(\mathrm { S } _ { \text {th } }\) and \(\mathrm { S } _ { h h }\). Give your answers to 3 significant figures.
52. Calculate the product moment correlation coefficient for this data.
53. State whether or not your value supports the use of a regression equation to predict the air temperature at different heights on this mountain. Give a reason for your answer.
54. Find the equation of the regression line of \(t\) on \(h\) giving your answer in the form \(t = a + b h\).
55. Interpret the value of \(b\).
56. Estimate the difference in air temperature between a height of 500 m and a height of 1000 m .
    2. The marks of a group of female students in a statistics test are summarised in Figure 1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e8afd947-55ac-424b-8db5-d5aa856ef4d7-083_412_723_331_390} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} An outlier is a mark that is
    either more than \(1.5 \times\) interquartile range above the upper quartile
    or more than \(1.5 \times\) interquartile range below the lower quartile.
57. On graph paper draw a box plot to represent the marks of the male students, indicating clearly any outliers.
58. Compare and contrast the marks of the male and the female students.
    (2) \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{(a) Write down the mark which is exceeded by \(75 \%\) of the female students.
    (1)
59. Write down the mark which is exceeded by \(75 \%\) of the female students.
    The marks of a group of male students in the same statistics test are summarised by the stem and leaf diagram below.}

    Mark	\(( 2 \mid 6\) means \(26 )\)	Totals
    1	4	\(( 1 )\)
    2	6	\(( 1 )\)
    3	447	\(( 3 )\)
    4	066778	\(( 6 )\)
    5	001113677	\(( 9 )\)
    6	223338	\(( 6 )\)
    7	008	\(( 3 )\)
    8	5	\(( 1 )\)
    9	0	\(( 1 )\)