2.04b Binomial distribution: as model B(n,p)

13 At a certain factory Christmas tree decorations are packed in boxes of 10 . The quality control manager collects a random sample of 100 boxes of decorations and records the number of decorations in each box which are damaged. His results are displayed in Fig. 13.1. \begin{table}[h] \end{table}

1. Calculate
   It is believed that the number of damaged decorations in a box of 10, \(X\), may be modelled by a binomial distribution such that \(\mathrm { X } \sim \mathrm { B } ( \mathrm { n } , \mathrm { p } )\).
2. State suitable values for \(n\) and \(p\).
3. Use the binomial model to complete the copy of Fig. 13.2 in the Printed Answer Booklet, giving your answers correct to \(\mathbf { 1 }\) decimal place. \begin{table}[h]

   Number of damaged decorations	0	1	2	3	4	5 or more
   Observed number of boxes	19	35	28	13	5	0
   Expected number of boxes

   \captionsetup{labelformat=empty} \caption{Fig. 13.2}
   \end{table}
4. Explain whether the model is a good fit for these data.

1. Gill buys a bag of logs to use in her stove. The lengths, \(l \mathrm {~cm}\), of the 88 logs in the bag are summarised in the table below.

A histogram is drawn to represent these data.
The bar representing logs with length \(27 < l \leqslant 30\) has a width of 1.5 cm and a height of 4 cm .

1. Calculate the width and height of the bar representing log lengths of \(20 < l \leqslant 25\)
2. Use linear interpolation to estimate the median of \(l\) The maximum length of log Gill can use in her stove is 26 cm .
   Gill estimates, using linear interpolation, that \(x\) logs from the bag will fit into her stove.
3. Show that \(x = 62\) Gill randomly selects 4 logs from the bag.
4. Using \(x = 62\), find the probability that all 4 logs will fit into her stove. The weights, \(W\) grams, of the logs in the bag are coded using \(y = 0.5 w - 255\) and summarised by $$n = 88 \quad \sum y = 924 \quad \sum y ^ { 2 } = 12862$$
5. Calculate
   1. the mean of \(W\)
   2. the variance of \(W\)

5. The discrete random variable \(X\) has probability function $$\mathrm { P } ( X = x ) = \begin{cases} k ( 2 - x ) , & x = 0,1,2 \\ k ( x - 2 ) , & x = 3 \\ 0 , & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant.

1. Show that \(k = 0.25\).
2. Find \(\mathrm { E } ( X )\) and show that \(\mathrm { E } \left( X ^ { 2 } \right) = 2.5\).
3. Find \(\operatorname { Var } ( 3 X - 2 )\). Two independent observations \(X _ { 1 }\) and \(X _ { 2 }\) are made of \(X\).
4. Show that \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } = 5 \right) = 0\).
5. Find the complete probability function for \(X _ { 1 } + X _ { 2 }\).
6. Find \(\mathrm { P } \left( 1.3 \leq X _ { 1 } + X _ { 2 } \leq 3.2 \right)\).

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

1. Show that \(k = 0.1\) Find
2. \(\mathrm { E } ( X )\)
3. \(\mathrm { E } \left( X ^ { 2 } \right)\)
4. \(\operatorname { Var } ( 2 - 5 X )\) Two independent observations \(X _ { 1 }\) and \(X _ { 2 }\) are made of \(X\).
5. Show that \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } = 4 \right) = 0.1\)
6. Complete the probability distribution table for \(X _ { 1 } + X _ { 2 }\)

   \(y\)	2	3	4	5	6	7	8
   \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } = y \right)\)	0.01	0.04	0.10		0.25	0.24

7. Find \(\mathrm { P } \left( 1.5 < X _ { 1 } + X _ { 2 } \leqslant 3.5 \right)\)

1. A discrete random variable \(X\) has the probability function

$$\mathrm { P } ( X = x ) = \begin{cases} k ( 1 - x ) ^ { 2 } & x = - 1,0,1 \text { and } 2 \\ 0 & \text { otherwise } \end{cases}$$

1. Show that \(k = \frac { 1 } { 6 }\)
2. Find \(\mathrm { E } ( X )\)
3. Show that \(\mathrm { E } \left( X ^ { 2 } \right) = \frac { 4 } { 3 }\)
4. Find \(\operatorname { Var } ( 1 - 3 X )\)

7. The score \(S\) when a spinner is spun has the following probability distribution.

1. Find \(\mathrm { E } ( S )\).
2. Show that \(\mathrm { E } \left( S ^ { 2 } \right) = 10.4\)
3. Hence find \(\operatorname { Var } ( S )\).
4. Find
   1. \(\mathrm { E } ( 5 S - 3 )\),
   2. \(\operatorname { Var } ( 5 S - 3 )\).
5. Find \(\mathrm { P } ( 5 S - 3 > S + 3 )\) The spinner is spun twice.
   The score from the first spin is \(S _ { 1 }\) and the score from the second spin is \(S _ { 2 }\) The random variables \(S _ { 1 }\) and \(S _ { 2 }\) are independent and the random variable \(X = S _ { 1 } \times S _ { 2 }\)
6. Show that \(\mathrm { P } \left( \left\{ S _ { 1 } = 1 \right\} \cap X < 5 \right) = 0.16\)
7. Find \(\mathrm { P } ( X < 5 )\).

4. A customer wishes to withdraw money from a cash machine. To do this it is necessary to type a PIN number into the machine. The customer is unsure of this number. If the wrong number is typed in, the customer can try again up to a maximum of four attempts in total. Attempts to type in the correct number are independent and the probability of success at each attempt is 0.6 .

1. Show that the probability that the customer types in the correct number at the third attempt is 0.096 .
   (2 marks)
   The random variable \(A\) represents the number of attempts made to type in the correct PIN number, regardless of whether or not the attempt is successful.
2. Find the probability distribution of \(A\).
3. Calculate the probability that the customer types in the correct number in four or fewer attempts.
4. Calculate \(\mathrm { E } ( A )\) and \(\operatorname { Var } ( A )\).
5. Find \(\mathrm { F } ( 1 + \mathrm { E } ( A ) )\).

1. The probability of a leaf cutting successfully taking root is 0.05

Find the probability that, in a batch of 10 randomly selected leaf cuttings, the number taking root will be

1. 1. exactly 1
   2. more than 2 A second random sample of 160 leaf cuttings is selected.
2. Using a suitable approximation, estimate the probability of at least 10 leaf cuttings taking root.
   

5. A school photocopier breaks down randomly at a rate of 15 times per year.

1. Find the probability that there will be exactly 3 breakdowns in the next month.
2. Show that the probability that there will be at least 2 breakdowns in the next month is 0.355 to 3 decimal places.
3. Find the probability of at least 2 breakdowns in each of the next 4 months. The teachers would like a new photocopier. The head teacher agrees to monitor the situation for the next 12 months. The head teacher decides he will buy a new photocopier if there is more than 1 month when the photocopier has at least 2 breakdowns.
4. Find the probability that the head teacher will buy a new photocopier.

6. The Headteacher of a school claims that \(30 \%\) of parents do not support a new curriculum. In a survey of 20 randomly selected parents, the number, \(X\), who do not support the new curriculum is recorded. Assuming that the Headteacher's claim is correct, find

1. the probability that \(X = 5\)
2. the mean and the standard deviation of \(X\) The Director of Studies believes that the proportion of parents who do not support the new curriculum is greater than \(30 \%\). Given that in the survey of 20 parents 8 do not support the new curriculum,
3. test, at the \(5 \%\) level of significance, the Director of Studies' belief. State your hypotheses clearly. The teachers believe that the sample in the original survey was biased and claim that only \(25 \%\) of the parents are in support of the new curriculum. A second random sample, of size \(2 n\), is taken and exactly half of this sample supports the new curriculum. A test is carried out at a 10\% level of significance of the teachers' belief using this sample of size \(2 n\) Using the hypotheses \(\mathrm { H } _ { 0 } : p = 0.25\) and \(\mathrm { H } _ { 1 } : p > 0.25\)
4. find the minimum value of \(n\) for which the outcome of the test is that the teachers' belief is rejected.

7. A multiple choice examination paper has \(n\) questions where \(n > 30\) Each question has 5 answers of which only 1 is correct. A pass on the paper is obtained by answering 30 or more questions correctly. The probability of obtaining a pass by randomly guessing the answer to each question should not exceed 0.0228 Use a normal approximation to work out the greatest number of questions that could be used.

The continuous random variable \(W\) has the normal distribution \(\mathrm { N } \left( 32,4 { } ^ { 2 } \right)\)

1. Write down the value of \(\mathrm { P } ( W = 36 )\) The discrete random variable \(X\) has the binomial distribution \(\mathrm { B } ( 20,0.45 )\)
2. Find \(\mathrm { P } ( X = 8 )\)
3. Find the probability that \(X\) lies within one standard deviation of its mean.
   

1. In the manufacture of cloth in a factory, defects occur randomly in the production process at a rate of 2 per \(5 \mathrm {~m} ^ { 2 }\)

The quality control manager randomly selects 12 pieces of cloth each of area \(15 \mathrm {~m} ^ { 2 }\).

1. Find the probability that exactly half of these 12 pieces of cloth will contain at most 7 defects. The factory introduces a new procedure to manufacture the cloth. After the introduction of this new procedure, the manager takes a random sample of \(25 \mathrm {~m} ^ { 2 }\) of cloth from the next batch produced to test if there has been any change in the rate of defects.
2. 1. Write down suitable hypotheses for this test.
   2. Describe a suitable test statistic that the manager should use.
   3. Explain what is meant by the critical region for this test.
3. Using a 5\% level of significance, find the critical region for this test. You should choose the largest critical region for which the probability in each tail is less than 2.5\%
4. Find the actual significance level for this test.

1. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by

$$f ( x ) = \begin{cases} \frac { 1 } { 16 } x ^ { 2 } & 1 \leqslant x < 3 \\ k ( 4 - x ) & 3 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$

1. Show that \(k = \frac { 11 } { 12 }\)
2. Sketch \(\mathrm { f } ( x )\) for \(1 \leqslant x \leqslant 4\)
3. Write down the mode of \(X\) Given that \(\mathrm { E } ( X ) = \frac { 25 } { 9 }\)
4. use algebraic integration to find \(\operatorname { Var } ( X )\), giving your answer to 3 significant figures. The cumulative distribution function of \(X\) is given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 1 \\ \frac { 1 } { 48 } \left( x ^ { 3 } + c \right) & 1 \leqslant x < 3 \\ \frac { 11 } { 12 } \left( 4 x - \frac { 1 } { 2 } x ^ { 2 } + d \right) & 3 \leqslant x \leqslant 4 \\ 1 & x > 4 \end{array} \right.$$
5. 1. Find the exact value of \(C\)
   2. Find the exact value of \(d\)
6. Calculate, to 3 significant figures, the upper quartile of \(X\)
   \includegraphics[max width=\textwidth, alt={}]{a814156d-6945-4601-9cae-d28d8ae0db1e-28_2632_1826_121_121}

1. During morning hours, employees arrive randomly at an office drinks dispenser at a rate of 2 every 10 minutes.

The number of employees arriving at the drinks dispenser is assumed to follow a Poisson distribution.

1. Find the probability that fewer than 5 employees arrive at the drinks dispenser during a 10-minute period one morning. During a 30 -minute period one morning, the probability that \(n\) employees arrive at the drinks dispenser is the same as the probability that \(n + 1\) employees arrive at the drinks dispenser.
2. Find the value of \(n\) During a 45-minute period one morning, the probability that between \(c\) and 12, inclusive, employees arrive at the drinks dispenser is 0.8546
3. Find the value of \(C\)
4. Find the probability that exactly 2 employees arrive at the drinks dispenser in exactly 4 of the 6 non-overlapping 10-minute intervals between 10 am and 11am one morning.

1. At a shop, past figures show that \(35 \%\) of customers pay by credit card. Following the shop's decision to no longer charge a fee for using a credit card, a random sample of 20 customers is taken and 11 are found to have paid by credit card.

Hadi believes that the proportion of customers paying by credit card is now greater than 35\%

1. Test Hadi's belief at the \(5 \%\) level of significance. State your hypotheses clearly. For a random sample of 20 customers,
2. show that 11 lies less than 2 standard deviations above the mean number of customers paying by credit card.
   You may assume that \(35 \%\) is the true proportion of customers who pay by credit card.
   

1. Jim farms oysters in a particular lake. He knows from past experience that \(5 \%\) of young oysters do not survive to be harvested.

In a random sample of 30 young oysters, the random variable \(X\) represents the number that do not survive to be harvested.

1. Write down a suitable model for the distribution of \(X\).
2. State an assumption that has been made for the model in part (a).
3. Find the probability that
   1. exactly 24 young oysters do survive to be harvested,
   2. at least 3 young oysters do not survive to be harvested. A second random sample, of 200 young oysters, is taken. The probability that at least \(n\) of these young oysters do not survive to be harvested is more than 0.8
4. Using a suitable approximation, find the maximum value of \(n\). Jim believes that the level of salt in the lake water has changed and it has altered the survival rate of his oysters. He takes a random sample of 25 young oysters and places them in the lake.
   When Jim harvests the oysters, he finds that 21 do survive to be harvested.
5. Use a suitable test, at the \(5 \%\) level of significance, to assess whether or not there is evidence that the proportion of oysters not surviving to be harvested is more than \(5 \%\). State your hypotheses clearly.

2. The distance, in metres, a novice tightrope artist, walking on a wire, walks before falling is modelled by the random variable \(W\) with cumulative distribution function $$\mathrm { F } ( w ) = \left\{ \begin{array} { c c } 0 & w < 0 \\ \frac { 1 } { 3 } \left( w - \frac { w ^ { 4 } } { 256 } \right) & 0 \leqslant w \leqslant 4 \\ 1 & w > 4 \end{array} \right.$$

1. Find the probability that a novice tightrope artist, walking on the wire, walks at least 3.5 metres before falling. A random sample of 30 novice tightrope artists is taken.
2. Find the probability that more than 1 of these novice tightrope artists, walking on the wire, walks at least 3.5 metres before falling. Given \(\mathrm { E } ( W ) = 1.6\)
3. use algebraic integration to find \(\operatorname { Var } ( W )\) DO NOT WRITEIN THIS AREA

3. The number of water fleas, in 100 ml of pond water, has a Poisson distribution with mean 7

1. Find the probability that a sample of 100 ml of the pond water does not contain exactly 4 water fleas. Aja collects 5 separate samples, each of 100 ml , of the pond water.
2. Find the probability that exactly 1 of these samples contains exactly 4 water fleas. Using a normal approximation, the probability that more than 3 water fleas will be found in a random sample of \(n \mathrm { ml }\) of the pond water is 0.9394 correct to 4 significant figures.
3. 1. Show that \(n - 1.55 \sqrt { \frac { n } { 0.07 } } - 50 = 0\)
   2. Hence find the value of \(n\) After the pond has been cleaned, the number of water fleas in a 100 ml random sample of the pond water is 15
4. Using a suitable test, at the \(1 \%\) level of significance, assess whether or not there is evidence that the number of water fleas per 100 ml of the pond water has increased. State your hypotheses clearly. \includegraphics[max width=\textwidth, alt={}, center]{f63c39df-cfc9-4a6b-838d-67613710b0ce-11_2255_50_314_34}
   

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1 A local pottery makes cups. The number of faulty cups made by the pottery in a week follows a Poisson distribution with a mean of 6 In a randomly chosen week, the probability that there will be at least \(x\) faulty cups made is 0.1528

1. Find the value of \(x\)
2. Use a normal approximation to find the probability that in 6 randomly chosen weeks the total number of faulty cups made is fewer than 32 A week is called a "poor week" if at least \(x\) faulty cups are made, where \(x\) is the value found in part (a).
3. Find the probability that in 50 randomly chosen weeks, more than 1 is a "poor week".

5 Applicants for a pilot training programme with a passenger airline are screened for colour blindness. Past records show that the proportion of applicants identified as colour blind is 0.045

1. Write down a suitable model for the distribution of the number of applicants identified as colour blind from a total of \(n\) applicants.
2. State one assumption necessary for this distribution to be a suitable model of this situation.
3. Using a suitable approximation, find the probability that exactly 5 out of 120 applicants are identified as colour blind.
4. Explain why the approximation that you used in part (c) is appropriate. Jaymini claims that 75\% of all applicants for this training programme go on to become pilots. From a random sample of 96 applicants for this training programme 67 go on to become pilots.
5. Using a suitable approximation, test Jaymini's claim at the \(5 \%\) level of significance. State your hypotheses clearly.

1. Superbounce is a manufacturer of tennis balls.

It knows from past records that 10\% of its tennis balls fail a bounce test.

1. Find the probability that from a random sample of 10 of these tennis balls
   1. at least 4 fail the bounce test
   2. more than 1 but fewer than 5 fail the bounce test. The managing director makes changes to the production process and claims that these changes will reduce the probability of its tennis balls failing the bounce test. After the changes were made a random sample of 50 of the tennis balls were tested and it was found that 2 failed the bounce test.
2. Test, at the \(5 \%\) significance level, whether or not this result supports the managing director's claim. In a second random sample of \(n\) tennis balls it was found that none failed the bounce test. As a result of this sample, the managing director's claim is supported at the 1\% significance level.
3. Find the smallest possible value of \(n\)

1. A company produces steel cable.

Defects in the steel cable produced by this company occur at random, at a constant rate of 1 defect per 16 metres. On one day the company produces a piece of steel cable 80 metres long.

1. Find the probability that there are at most 5 defects in this piece of steel cable. The company produces a piece of steel cable 80 metres long on each of the next 4 days.
2. Find the probability that fewer than 2 of these 4 pieces of steel cable contain at most 5 defects. The following week the company produces a piece of steel cable \(x\) metres long.
   Using a normal approximation, the probability that this piece of steel cable has fewer than 26 defects is 0.5398
3. Find the value of \(x\)

1. The manager of a supermarket is investigating the number of complaints per day received from customers.

A random sample of 180 days is taken and the results are shown in the table below.

1. Calculate the mean and the variance of these data.
2. Explain why the results in part (a) suggest that a Poisson distribution may be a suitable model for the number of complaints per day. The manager uses a Poisson distribution with mean 3 to model the number of complaints per day.
3. For a randomly selected day find, using the manager's model, the probability that there are
   1. at least 3 complaints,
   2. more than 4 complaints but less than 8 complaints. A week consists of 7 consecutive days.
4. Using the manager's model and a suitable approximation, show that the probability that there are less than 19 complaints in a randomly selected week is 0.29 to 2 decimal places.
   Show your working clearly.
   (Solutions relying on calculator technology are not acceptable.) A period of 13 weeks is selected at random.
5. Find the probability that in this period there are exactly 5 weeks that have less than 19 complaints.
   Show your working clearly.

1. The length of pregnancy for a randomly selected pregnant sheep is \(D\) days where

$$D \sim \mathrm {~N} \left( 112.4 , \sigma ^ { 2 } \right)$$ Given that 5\% of pregnant sheep have a length of pregnancy of less than 108 days,

1. find the value of \(\sigma\) Qiang selects 25 pregnant sheep at random from a large flock.
2. Find the probability that more than 3 of these pregnant sheep have a length of pregnancy of less than 108 days. Charlie takes 200 random samples of 25 pregnant sheep.
3. Use a Poisson approximation to estimate the probability that at least 2 of the samples have more than 3 pregnant sheep with a length of pregnancy of less than 108 days.