2.04e Normal distribution: as model N(mu, sigma^2)

1. The weight of coffee in glass jars labelled 100 g is normally distributed with mean 101.80 g and standard deviation 0.72 g . The weight of an empty glass jar is normally distributed with mean 260.00 g and standard deviation 5.45 g . The weight of a glass jar is independent of the weight of the coffee it contains.

Find the probability that a randomly selected jar weighs less than 266 g and contains less than 100 g of coffee. Give your answer to 2 significant figures.
(8 marks)

3. Cooking sauces are sold in jars containing a stated weight of 500 g of sauce The jars are filled by a machine. The actual weight of sauce in each jar is normally distributed with mean 505 g and standard deviation 10 g .

1. 1. Find the probability of a jar containing less than the stated weight.
   2. In a box of 30 jars, find the expected number of jars containing less than the stated weight. The mean weight of sauce is changed so that \(1 \%\) of the jars contain less than the stated weight. The standard deviation stays the same.
2. Find the new mean weight of sauce.

3. The random variable \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\). It is known that $$\mathrm { P } ( X \leq 66 ) = 0.0359 \text { and } \mathrm { P } ( X \geq 81 ) = 0.1151 .$$

1. In the space below, give a clearly labelled sketch to represent these probabilities on a Normal curve.
2. 1. Show that the value of \(\sigma\) is 5 .
   2. Find the value of \(\mu\).
3. Find \(\mathrm { P } ( 69 \leq X \leq 83 )\).

6. Students in Mr Brawn's exercise class have to do press-ups and sit-ups. The number of press-ups \(x\) and the number of sit-ups \(y\) done by a random sample of 8 students are summarised below. $$\begin{array} { l l } \Sigma x = 272 , & \Sigma x ^ { 2 } = 10164 , \quad \Sigma x y = 11222 , \\ \Sigma y = 320 , & \Sigma y ^ { 2 } = 13464 . \end{array}$$

1. Evaluate \(S _ { x x } , S _ { y y }\) and \(S _ { x y }\).
2. Calculate, to 3 decimal places, the product moment correlation coefficient between \(x\) and \(y\).
3. Give an interpretation of your coefficient.
4. Calculate the mean and the standard deviation of the number of press-ups done by these students. Mr Brawn assumes that the number of press-ups that can be done by any student can be modelled by a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). Assuming that \(\mu\) and \(\sigma\) take the same values as those calculated in part (d),
5. find the value of \(a\) such that \(\mathrm { P } ( \mu - a < X < \mu + a ) = 0.95\).
6. Comment on Mr Brawn's assumption of normality.

1. The random variable \(Y \sim \mathrm {~B} ( n , p )\).

Using a normal approximation the probability that \(Y\) is at least 65 is 0.2266 and the probability that \(Y\) is more than 52 is 0.8944 Find the value of \(n\) and the value of \(p\).

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. 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.

4. Pieces of ribbon are cut to length \(L \mathrm {~cm}\) where \(L \sim \mathrm {~N} \left( \mu , 0.5 ^ { 2 } \right)\)

1. Given that \(30 \%\) of the pieces of ribbon have length more than 100 cm , find the value of \(\mu\) to the nearest 0.1 cm . John selects 12 pieces of ribbon at random.
2. Find the probability that fewer than 3 of these pieces of ribbon have length more than 100 cm . Aditi selects 400 pieces of ribbon at random.
3. Using a suitable approximation, find the probability that more than 127 of these pieces of ribbon will have length more than 100 cm .

5 A receptionist receives incoming telephone calls and should connect them to the appropriate department. The probability of them being connected to the wrong department on the first attempt is 0.05 A random sample of 8 calls is taken.

1. Find the probability that at least 2 of these calls are connected to the wrong department on the first attempt. The receptionist receives 1000 calls each day.
2. Use a Poisson approximation to find the probability that exactly 45 callers are connected to the wrong department on the first attempt in a day. The total time, \(T\) seconds, taken for a call to be answered by a department has a continuous uniform distribution over the interval [10,50]
3. Find \(\mathrm { P } ( T > 16 )\) The number of calls the receptionist receives in a one-minute interval is modelled by a Poisson distribution with mean 6 The receptionist receives a call from Jia and tries to connect it to the right department.
4. Find the probability that in the next 40 seconds Jia's call is answered by the right department on the first attempt and the receptionist has received no other calls.

According to an electric company, power failures occur randomly at a rate of \(\lambda\) every 10 weeks, \(1 < \lambda < 10\)

1. Write down an expression in terms of \(\lambda\) for the probability that there are fewer than 2 power failures in a randomly selected 10 week period.
2. Write down an expression in terms of \(\lambda\) for the probability that there is exactly 1 power failure in a randomly selected 5 week period. Over a 100 week period, the probability, using a normal approximation, that fewer than 15 power failures occur is 0.0179 (to 3 significant figures).
3. 1. Justify the use of a normal approximation.
   2. Find the value of \(\lambda\). Show each stage of your working clearly.
      

1. A shop sells rods of nominal length 200 cm . The rods are bought from a manufacturer who uses a machine to cut rods of length \(L \mathrm {~cm}\), where \(L \sim \mathrm {~N} \left( \mu , 0.2 ^ { 2 } \right)\)

The value of \(\mu\) is such that there is only a \(5 \%\) chance that a rod, selected at random from those supplied to the shop, will have length less than 200 cm .

1. Find the value of \(\mu\) to one decimal place. A customer buys a random sample of 8 of these rods.
2. Find the probability that at least 3 of these rods will have length less than 200 cm . Another customer buys a random sample of 60 of these rods.
3. Using a suitable approximation, find the probability that more than 5 of these rods will have length less than 200 cm .
   

6. A fair 6 -sided die is thrown \(n\) times. The number of sixes, \(X\), is recorded. Using a normal approximation, \(\mathrm { P } ( X < 50 ) = 0.0082\) correct to 4 decimal places. Find the value of \(n\).
(10)

1. At a cafe, customers ordering hot drinks order either tea or coffee.

Of all customers ordering hot drinks, \(80 \%\) order tea and \(20 \%\) order coffee. Of those who order tea, \(35 \%\) take sugar and of those who order coffee \(60 \%\) take sugar.

1. A random sample of 12 customers ordering hot drinks is selected. Find the probability that fewer than 3 of these customers order coffee.
2. 1. A randomly selected customer who orders a hot drink is chosen. Show that the probability that the customer takes sugar is 0.4
   2. Write down the distribution for the number of customers who take sugar from a random sample of \(n\) customers ordering hot drinks.
3. A random sample of 10 customers ordering hot drinks is selected.
   1. Find the probability that exactly 4 of these 10 customers take sugar.
   2. Given that at least 3 of these 10 customers take sugar, find the probability that no more than 6 of these 10 customers take sugar.
4. In a random sample of 150 customers ordering hot drinks, find, using a suitable approximation, the probability that at least half of them take sugar.

One side of a square is measured to the nearest centimetre and this measurement is multiplied by 4 to estimate the perimeter of the square. The random variable, \(W \mathrm {~cm}\), represents the estimated perimeter of the square minus the true perimeter of the square. \(W\) is uniformly distributed over the interval \([ a , b ]\)

1. Explain why \(a = - 2\) and \(b = 2\) The standard deviation of \(W\) is \(\sigma\)
2. 1. Find \(\sigma\)
   2. Find the probability that the estimated perimeter of the square is within \(\sigma\) of the true perimeter of the square. One side of each of 100 squares are now measured. Using a suitable approximation,
3. find the probability that \(W\) is greater than 1.9 for at least 5 of these squares.

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5. A farmer noticed that some of the eggs laid by his hens had double yolks. He estimated the probability of this happening to be 0.05 . Eggs are packed in boxes of 12 . Find the probability that in a box, the number of eggs with double yolks will be

1. exactly one,
2. more than three. A customer bought three boxes.
3. Find the probability that only 2 of the boxes contained exactly 1 egg with a double yolk. The farmer delivered 10 boxes to a local shop.
4. Using a suitable approximation, find the probability that the delivery contained at least 9 eggs with double yolks. The weight of an individual egg can be modelled by a normal distribution with mean 65 g and standard deviation 2.4 g .
5. Find the probability that a randomly chosen egg weighs more than 68 g .

1. The random variables \(R , S\) and \(T\) are distributed as follows

$$R \sim \mathrm {~B} ( 15,0.3 ) , \quad S \sim \mathrm { Po } ( 7.5 ) , \quad T \sim \mathrm {~N} \left( 8,2 ^ { 2 } \right) .$$ Find

1. \(\mathrm { P } ( R = 5 )\),
2. \(\mathrm { P } ( S = 5 )\),
3. \(\mathrm { P } ( T = 5 )\).
   

4. In an experiment, there are 250 trials and each trial results in a success or a failure.

1. Write down two other conditions needed to make this into a binomial experiment. It is claimed that \(10 \%\) of students can tell the difference between two brands of baked beans. In a random sample of 250 students, 40 of them were able to distinguish the difference between the two brands.
2. Using a normal approximation, test at the \(1 \%\) level of significance whether or not the claim is justified. Use a one-tailed test.
3. Comment on the acceptability of the assumptions you needed to carry out the test.

5. From company records, a manager knows that the probability that a defective article is produced by a particular production line is 0.032 . A random sample of 10 articles is selected from the production line.

1. Find the probability that exactly 2 of them are defective. On another occasion, a random sample of 100 articles is taken.
2. Using a suitable approximation, find the probability that fewer than 4 of them are defective. At a later date, a random sample of 1000 is taken.
3. Using a suitable approximation, find the probability that more than 42 are defective.
   (6)

7. A teacher thinks that \(20 \%\) of the pupils in a school read the Deano comic regularly. He chooses 20 pupils at random and finds 9 of them read the Deano.

1. 1. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the percentage of pupils that read the Deano is different from 20\%. State your hypotheses clearly.
   2. State all the possible numbers of pupils that read the Deano from a sample of size 20 that will make the test in part (a)(i) significant at the \(5 \%\) level.
      (9) The teacher takes another 4 random samples of size 20 and they contain 1, 3, 1 and 4 pupils that read the Deano.
2. By combining all 5 samples and using a suitable approximation test, at the \(5 \%\) level of significance, whether or not this provides evidence that the percentage of pupils in the school that read the Deano is different from 20\%.
3. Comment on your results for the tests in part (a) and part (b).

3. For a particular type of plant \(45 \%\) have white flowers and the remainder have coloured flowers. Gardenmania sells plants in batches of 12. A batch is selected at random. Calculate the probability that this batch contains

1. exactly 5 plants with white flowers,
2. more plants with white flowers than coloured ones. Gardenmania takes a random sample of 10 batches of plants.
3. Find the probability that exactly 3 of these batches contain more plants with white flowers than coloured ones. Due to an increasing demand for these plants by large companies, Gardenmania decides to sell them in batches of 50 .
4. Use a suitable approximation to calculate the probability that a batch of 50 plants contains more than 25 plants with white flowers.

4.

1. State the condition under which the normal distribution may be used as an approximation to the Poisson distribution.
2. Explain why a continuity correction must be incorporated when using the normal distribution as an approximation to the Poisson distribution. A company has yachts that can only be hired for a week at a time. All hiring starts on a Saturday.
   During the winter the mean number of yachts hired per week is 5 .
3. Calculate the probability that fewer than 3 yachts are hired on a particular Saturday in winter. During the summer the mean number of yachts hired per week increases to 25 . The company has only 30 yachts for hire.
4. Using a suitable approximation find the probability that the demand for yachts cannot be met on a particular Saturday in the summer. In the summer there are 16 Saturdays on which a yacht can be hired.
5. Estimate the number of Saturdays in the summer that the company will not be able to meet the demand for yachts.

5. The continuous random variable \(X\) is uniformly distributed over the interval \(\alpha < x < \beta\).

1. Write down the probability density function of \(X\), for all \(x\).
2. Given that \(\mathrm { E } ( X ) = 2\) and \(\mathrm { P } ( X < 3 ) = \frac { 5 } { 8 }\) find the value of \(\alpha\) and the value of \(\beta\). A gardener has wire cutters and a piece of wire 150 cm long which has a ring attached at one end. The gardener cuts the wire, at a randomly chosen point, into 2 pieces. The length, in cm, of the piece of wire with the ring on it is represented by the random variable \(X\). Find
3. \(\mathrm { E } ( X )\),
4. the standard deviation of \(X\),
5. the probability that the shorter piece of wire is at most 30 cm long.

6. The probability that a sunflower plant grows over 1.5 metres high is 0.25 . A random sample of 40 sunflower plants is taken and each sunflower plant is measured and its height recorded.

1. Find the probability that the number of sunflower plants over 1.5 m high is between 8 and 13 (inclusive) using
   1. a Poisson approximation,
   2. a Normal approximation.
2. Write down which of the approximations used in part (a) is the most accurate estimate of the probability. You must give a reason for your answer.

1. A café serves breakfast every morning. Customers arrive for breakfast at random at a rate of 1 every 6 minutes.

Find the probability that

1. fewer than 9 customers arrive for breakfast on a Monday morning between 10 am and 11 am. The café serves breakfast every day between 8 am and 12 noon.
2. Using a suitable approximation, estimate the probability that more than 50 customers arrive for breakfast next Tuesday.
   

5. The time taken for a randomly selected person to complete a test is \(M\) minutes, where \(M \sim \mathrm {~N} \left( 14 , \sigma ^ { 2 } \right)\) Given that \(10 \%\) of people take less than 12 minutes to complete the test,

1. find the value of \(\sigma\) Graham selects 15 people at random.
2. Find the probability that fewer than 2 of these people will take less than 12 minutes to complete the test. Jovanna takes a random sample of \(n\) people. Using a normal approximation, the probability that fewer than 9 of these \(n\) people will take less than 12 minutes to complete the test is 0.3085 to 4 decimal places.
3. Find the value of \(n\).