5.02c Linear coding: effects on mean and variance

1. A random sample of 75 packets of seeds is selected from a production line. Each packet contains 12 seeds. The seeds are planted and the number of seeds that germinate from each packet is recorded. The results are as follows.

1. Show that the probability of a randomly selected seed from this sample germinating is 0.82 A gardener suggests that a binomial distribution can be used to model the number of seeds that germinate from a packet of 12 seeds. She uses a binomial distribution with the estimated probability 0.82 of a seed germinating. Some of the calculated expected frequencies are shown in the table below.

   Number of seeds that germinate from each packet	6 or fewer	7	8	9	10	11	12
   Expected frequency	\(s\)	2.80	7.97	\(r\)	22.04	18.26	6.93

2. Calculate the value of \(r\) and the value of \(s\), giving your answers correct to 2 decimal places.
3. Test, at the \(10 \%\) level of significance, whether or not these data suggest that the binomial distribution is a suitable model for the number of seeds that germinate from a packet of 12 seeds. State your hypotheses clearly and show your working.

1. A researcher is looking into the effectiveness of a new medicine for the relief of symptoms. He collects random samples of 8 people who are taking the medicine from each of 50 different medical practices. The number of people who say that the medicine is a success, in each sample, is recorded. The results are summarised in the table below.

The researcher decides to model this data using a binomial distribution.

1. State two necessary assumptions that the researcher made in order to use this model.
2. Show that the mean number of successes per sample is 3.54 He decides to use this mean to calculate expected frequencies. The results are shown in the table below.

   Number of successes	0	1	2	3	4	5	6	7	8
   Expected frequency	0.47	2.96	8.23	13.07	\(f\)	8.23	3.27	0.74	\(g\)

3. Calculate the value of \(f\) and the value of \(g\). Give your answers to 2 decimal places.
4. Stating your hypotheses clearly, test at the \(10 \%\) level of significance, whether or not the binomial distribution is a suitable model for the number of successes in samples of 8 people.

6. A total of 100 random samples of 6 items are selected from a production line in a factory and the number of defective items in each sample is recorded. The results are summarised in the table below.

1. Show that the mean number of defective items per sample is 2.91 A factory manager suggests that the data can be modelled by a binomial distribution with \(n = 6\). He uses the mean from the sample above and calculates expected frequencies as shown in the table below.

   Number of defective items	0	1	2	3	4	5	6
   Expected frequency	1.87	10.54	24.82	\(a\)	22.01	8.29	\(b\)

2. Calculate the value of \(a\) and the value of \(b\) giving your answers to 2 decimal places.
3. Test, at the \(5 \%\) level, whether or not the binomial distribution is a suitable model for the number of defective items in samples of 6 items. State your hypotheses clearly.

6. In a survey for a computer magazine, the times \(t\) seconds taken by eight laser printers to print a page of text were compared with the prices \(\pounds p\) of the printers. The data were coded using the equations \(x = t - 10\) and \(y = p - 150\), and it was found that $$\sum x = 42 \cdot 4 , \quad \sum x ^ { 2 } = 314 \cdot 5 , \quad \sum y = 560 , \quad \sum y ^ { 2 } = 60600 , \quad \sum x y = 1592 .$$

1. Find the mean time and the mean price for the eight printers.
2. Find the variance of the times.
3. Find the equation of the regression line of \(p\) on \(t\).
4. Estimate the price of a printer which takes 11.3 seconds to print the page.

2. The discrete random variable \(Q\) has the following probability distribution.

1. Write down the name of this distribution. The discrete random variable \(R\) has the following probability distribution.

   \(r\)	14	24	34	44	54
   \(\mathrm { P } ( R = r )\)	\(\frac { 1 } { 5 }\)	\(\frac { 1 } { 5 }\)	\(\frac { 1 } { 5 }\)	\(\frac { 1 } { 5 }\)	\(\frac { 1 } { 5 }\)

2. State the relationship between \(R\) and \(Q\) in the form \(R = a Q + b\). Given that \(\mathrm { E } ( Q ) = 3\) and \(\operatorname { Var } ( Q ) = 2\),
3. find \(\mathrm { E } ( R )\) and \(\operatorname { Var } ( R )\).

5. The discrete random variable \(Y\) has the following cumulative distribution function.

1. Write down the probability distribution of \(Y\).
2. Find \(\mathrm { P } ( 1 \leq Y < 3 )\).
3. Show that \(\mathrm { E } ( Y ) = 2.7\)
4. Find \(\mathrm { E } ( 2 Y + 4 )\).
5. Find \(\operatorname { Var } ( Y )\).

1. The discrete random variable \(Y\) has the following probability distribution.

Find

1. \(\mathrm { F } ( 0.5 )\),
2. \(\mathrm { P } \left( { } ^ { - } 1 < Y < 1.9 \right)\),
3. \(\mathrm { E } ( Y )\),
4. \(\mathrm { E } ( 3 Y - 1 )\).
   

3. A magazine collected data on the total cost of the reception at each of a random sample of 80 weddings. The data is grouped and coded using \(y = \frac { C - 3250 } { 250 }\), where \(C\) is the mid-point in pounds of each class, giving \(\sum f y = 37\) and \(\sum f y ^ { 2 } = 2317\).

1. Using these values, calculate estimates of the mean and standard deviation of the cost of the receptions in the sample.
2. Explain why your answers to part (a) are only estimates. The median of the data was \(\pounds 3050\).
3. Comment on the skewness of the data and suggest a reason for it.

3. The random variable \(X\) is such that $$\mathrm { E } ( X ) = a \text { and } \operatorname { Var } ( X ) = b$$ Find expressions in terms of \(a\) and \(b\) for

1. \(\mathrm { E } ( 2 X + 3 )\),
2. \(\quad \operatorname { Var } ( 2 X + 3 )\),
3. \(\mathrm { E } \left( X ^ { 2 } \right)\).
4. Show that $$\mathrm { E } \left[ ( X + 1 ) ^ { 2 } \right] = ( a + 1 ) ^ { 2 } + b$$

5 The Globe Express agency organises trips to the theatre. The cost, \(\pounds X\), of these trips can be modelled by the following probability distribution:

1. Calculate the mean and standard deviation of \(X\).
2. For special celebrity charity performances, Globe Express increases the cost of the trips to \(\pounds Y\), where $$Y = 10 X + 250$$ Determine the mean and standard deviation of \(Y\).

4 The number of fish, \(X\), caught by Pearl when she goes fishing can be modelled by the following discrete probability distribution:

1. Find the value of \(k\).
2. Find:
   1. \(\mathrm { E } ( X )\);
   2. \(\operatorname { Var } ( X )\).
3. When Pearl sells her fish, she earns a profit, in pounds, given by $$Y = 5 X + 2$$ Find:
   1. \(\mathrm { E } ( Y )\);
   2. the standard deviation of \(Y\).

5

1. In a remote African village, it is known that 70 per cent of the villagers have a particular blood disorder. A medical research student selects 25 of the villagers at random. Using a binomial distribution, calculate the probability that more than 15 of these 25 villagers have this blood disorder.
2. 1. In towns and cities in Asia, the number of people who have this blood disorder may be modelled by a Poisson distribution with a mean of 2.6 per 100000 people. A town in Asia with a population of 100000 is selected. Determine the probability that at most 5 people have this blood disorder.
   2. In towns and cities in South America, the number of people who have this blood disorder may be modelled by a Poisson distribution with a mean of 49 per million people. A town in South America with a population of 100000 is selected. Calculate the probability that exactly 10 people have this blood disorder.
   3. The random variable \(T\) denotes the total number of people in the two selected towns who have this blood disorder. Write down the distribution of \(T\) and hence determine \(\mathrm { P } ( T > 16 )\).

5 In a computer game, players try to collect five treasures. The number of treasures that Isaac collects in one play of the game is represented by the discrete random variable \(X\). The probability distribution of \(X\) is defined by $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } \frac { 1 } { x + 2 } & x = 1,2,3,4 \\ k & x = 5 \\ 0 & \text { otherwise } \end{array} \right.$$

1. 1. Show that \(k = \frac { 1 } { 20 }\).
   2. Calculate the value of \(\mathrm { E } ( X )\).
   3. Show that \(\operatorname { Var } ( X ) = 1.5275\).
   4. Find the probability that Isaac collects more than 2 treasures.
2. The number of points that Isaac scores for collecting treasures is \(Y\) where $$Y = 100 X - 50$$ Calculate the mean and the standard deviation of \(Y\).

3. In an old computer game a white square representing a ball appears at random at the top of the playing area, which is 24 cm wide, and moves down the screen. The continuous random variable \(X\) represents the distance, in centimetres, of the dot from the left-hand edge of the screen when it appears. The distribution of \(X\) is rectangular over the interval [4,28].

1. Find the mean and variance of \(X\).
2. Find \(\mathrm { P } ( | X - 16 | < 3 )\). During a single game, a player receives 12 "balls".
3. Find the probability that the ball appears within 3 cm of the middle of the top edge of the playing area more than four times in a single game.
   (3 marks)

5. Four coins are flipped together and the random variable \(H\) represents the number of heads obtained. Assuming that the coins are fair,

1. suggest with reasons a suitable distribution for modelling \(H\) and give the value of any parameters needed,
2. show that the probability of obtaining more heads than tails is \(\frac { 5 } { 16 }\). The four coins are flipped 5 times and more heads are obtained than tails 4 times.
3. Stating your hypotheses clearly, test at the \(5 \%\) level of significance whether or not there is evidence of the probability of getting more heads than tails being more than \(\frac { 5 } { 16 }\). Given that the four coins are all biased such that the chance of each one showing a head is 50\% more than the chance of it showing a tail,
4. find the probability of obtaining more heads than tails when the four coins are flipped together.

4 A manufacturer produces three models of washing machine: basic, standard and deluxe. An analysis of warranty records shows that \(25 \%\) of faults are on basic machines, \(60 \%\) are on standard machines and 15\% are on deluxe machines. For basic machines, 30\% of faults reported during the warranty period are electrical, \(50 \%\) are mechanical and \(20 \%\) are water-related. For standard machines, 40\% of faults reported during the warranty period are electrical, \(45 \%\) are mechanical and 15\% are water-related. For deluxe machines, \(55 \%\) of faults reported during the warranty period are electrical, \(35 \%\) are mechanical and \(10 \%\) are water-related.

1. Draw a tree diagram to represent the above information.
2. Hence, or otherwise, determine the probability that a fault reported during the warranty period:
   1. is electrical;
   2. is on a deluxe machine, given that it is electrical.
3. A random sample of 10 electrical faults reported during the warranty period is selected. Calculate the probability that exactly 4 of them are on deluxe machines.

6. The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) are each distributed \(\mathrm { B } ( n , p )\), where \(n > 1\) An unbiased estimator for \(p\) is given by $$\hat { p } = \frac { a X _ { 1 } + b X _ { 2 } } { n }$$ where \(a\) and \(b\) are constants.
[0pt] [You may assume that if \(X _ { 1 }\) and \(X _ { 2 }\) are independent then \(\mathrm { E } \left( X _ { 1 } X _ { 2 } \right) = \mathrm { E } \left( X _ { 1 } \right) \mathrm { E } \left( X _ { 2 } \right)\) ]

1. Show that \(a + b = 1\)
2. Show that \(\operatorname { Var } ( \hat { p } ) = \frac { \left( 2 a ^ { 2 } - 2 a + 1 \right) p ( 1 - p ) } { n }\)
3. Hence, justifying your answer, determine the value of \(a\) and the value of \(b\) for which \(\hat { p }\) has minimum variance.
4. 1. Show that \(\hat { p } ^ { 2 }\) is a biased estimator for \(p ^ { 2 }\)
   2. Show that the bias \(\rightarrow 0\) as \(n \rightarrow \infty\)
5. By considering \(\mathrm { E } \left[ X _ { 1 } \left( X _ { 1 } - 1 \right) \right]\) find an unbiased estimator for \(p ^ { 2 }\)

3. A certain vaccine is known to be only \(70 \%\) effective against a particular virus; thus \(30 \%\) of those vaccinated will actually catch the virus. In order to test whether or not a new and more expensive vaccine provides better protection against the same virus, a random sample of 30 people were chosen and given the new vaccine. If fewer than 6 people contracted the virus the new vaccine would be considered more effective than the current one.

1. Write down suitable hypotheses for this test.
2. Find the probability of making a Type I error.
3. Find the power of this test if the new vaccine is
   1. \(80 \%\) effective,
   2. \(90 \%\) effective. An independent research organisation decided to test the new vaccine on a random sample of 50 people to see if it could be considered more than \(70 \%\) effective. They required the probability of a Type I error to be as close as possible to 0.05 .
4. Find the critical region for this test.
5. State the size of this critical region.
6. Find the power of this test if the new vaccine is
   1. \(80 \%\) effective,
   2. \(90 \%\) effective.
7. Give one advantage and one disadvantage of the second test.

6. A statistics student is trying to estimate the probability, \(p\), of rolling a 6 with a particular die. The die is rolled 10 times and the random variable \(X _ { 1 }\) represents the number of sixes obtained. The random variable \(R _ { 1 } = \frac { X _ { 1 } } { 10 }\) is proposed as an estimator of \(p\).

1. Show that \(R _ { 1 }\) is an unbiased estimator of \(p\). The student decided to roll the die again \(n\) times ( \(n > 10\) ) and the random variable \(X _ { 2 }\) represents the number of sixes in these \(n\) rolls. The random variable \(R _ { 2 } = \frac { X _ { 2 } } { n }\) and the random variable \(Y = \frac { 1 } { 2 } \left( R _ { 1 } + R _ { 2 } \right)\).
2. Show that both \(R _ { 2 }\) and \(Y\) are unbiased estimators of \(p\).
3. Find \(\operatorname { Var } \left( R _ { 2 } \right)\) and \(\operatorname { Var } ( Y )\).
4. State giving a reason which of the 3 estimators \(R _ { 1 } , R _ { 2 }\) and \(Y\) are consistent estimators of \(p\).
5. For the case \(n = 20\) state, giving a reason, which of the 3 estimators \(R _ { 1 } , R _ { 2 }\) and \(Y\) you would recommend. The student's teacher pointed out that a better estimator could be found based on the random variable \(X _ { 1 } + X _ { 2 }\).
6. Find a suitable estimator and explain why it is better than \(R _ { 1 } , R _ { 2 }\) and \(Y\). END

1 The discrete random variable \(X\) has probability distribution defined by $$\mathrm { P } ( X = r ) = k \left( r ^ { 2 } + 3 r \right) \text { for } r = 1,2,3,4,5 \text {, where } k \text { is a constant. }$$

1. Complete the table below, using the copy in the Printed Answer Booklet giving the probabilities in terms of \(k\).

   \(r\)	1	2	3	4	5
   \(\mathrm { P } ( X = r )\)	\(4 k\)	\(10 k\)

2. Show that the value of \(k\) is 0.01 .
3. Draw a graph to illustrate the distribution.
4. Describe the shape of the distribution.
5. Find each of the following.

3 A fair 8 -sided dice has faces labelled 10, 20, 30, ..., 80 .

1. State the distribution of the score when the dice is rolled once.
2. Write down the probability that, when the dice is rolled once, the score is at least 40 .
3. The dice is rolled three times.
   1. Find the variance of the total score obtained.
   2. Find the probability that on one of the rolls the score is less than 30 , on another it is between 30 and 50 inclusive and on the other it is greater than 50 .

1 A fair five-sided spinner has sectors labelled 1, 2, 3, 4, 5. In a game at a stall at a charity event, the spinner is spun twice. The random variable \(X\) represents the lower of the two scores. The probability distribution of \(X\) is given by the formula \(\mathrm { P } ( \mathrm { X } = \mathrm { r } ) = \mathrm { k } ( 11 - 2 \mathrm { r } )\) for \(r = 1,2,3,4,5\),
where \(k\) is a constant.

1. Complete the copy of this table in the Printed Answer Booklet.

   \(r\)	1	2	3	4	5
   \(\mathrm { P } ( X = r )\)		\(7 k\)		\(3 k\)

2. Determine the value of \(k\).
3. Find each of the following.
   Given that the average profit that the stall-holder makes on one game is 25 pence, find the value of \(C\).

1 Ryan has 6 one-pound coins and 4 two-pound coins. Ryan decides to select 3 of these coins at random to donate to a charity. The total value, in pounds, of these 3 coins is denoted by the random variable \(X\).

1. Show that \(\mathrm { P } ( X = 3 ) = \frac { 1 } { 6 }\). The table below shows the probability distribution of \(X\).

   \(r\)	3	4	5	6
   \(\mathrm { P } ( \mathrm { X } = \mathrm { r } )\)	\(\frac { 1 } { 6 }\)	\(\frac { 1 } { 2 }\)	\(\frac { 3 } { 10 }\)	\(\frac { 1 } { 30 }\)

2. Draw a graph to illustrate the distribution.
3. In this question you must show detailed reasoning. Find each of the following.
   Ryan's friend Sasha decides to give the same amount as Ryan does to the charity plus an extra three pounds. The random variable \(Y\) represents the total amount of money, in pounds, given by Ryan and Sasha.
4. Determine each of the following.

1 The probability distribution for a discrete random variable \(X\) is given in the table below.

1. Find the value of \(c\).
2. Find the value of each of the following.
   The random variable \(Y\) is defined by \(Y = 2 X - 3\).
3. Find the value of each of the following.