2.04f Find normal probabilities: Z transformation

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 \(\sigma\).

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

$$\mathrm { P } ( X = x ) = \frac { 1 } { 10 } \quad x = 1,2,3 , \ldots 10$$

1. Write down the name given to this distribution.
2. Write down the value of
   1. \(\mathrm { P } ( X = 10 )\)
   2. \(\mathrm { P } ( X < 10 )\) The continuous random variable \(Y\) has the normal distribution \(\mathrm { N } \left( 10,2 ^ { 2 } \right)\)
3. Write down the value of
   1. \(\mathrm { P } ( Y = 10 )\)
   2. \(\mathrm { P } ( Y < 10 )\)

6. The time taken, in minutes, by children to complete a mathematical puzzle is assumed to be normally distributed with mean \(\mu\) and standard deviation \(\sigma\). The puzzle can be completed in less than 24 minutes by \(80 \%\) of the children. For \(5 \%\) of the children it takes more than 28 minutes to complete the puzzle.

1. Show this information on the Normal curve below.
2. Write down the percentage of children who take between 24 minutes and 28 minutes to complete the puzzle.
3. 1. Find two equations in \(\mu\) and \(\sigma\).
   2. Hence find, to 3 significant figures, the value of \(\mu\) and the value of \(\sigma\). A child is selected at random.
4. Find the probability that the child takes less than 12 minutes to complete the puzzle. \includegraphics[max width=\textwidth, alt={}, center]{ca8418eb-4d35-40f4-af40-77503327ae52-11_314_1255_1375_356}

7. The heights of adult females are normally distributed with mean 160 cm and standard deviation 8 cm .

1. Find the probability that a randomly selected adult female has a height greater than 170 cm . Any adult female whose height is greater than 170 cm is defined as tall. An adult female is chosen at random. Given that she is tall,
2. find the probability that she has a height greater than 180 cm . Half of tall adult females have a height greater than \(h \mathrm {~cm}\).
3. Find the value of \(h\).

The random variable \(Z \sim \mathrm {~N} ( 0,1 )\) \(A\) is the event \(Z > 1.1\) \(B\) is the event \(Z > - 1.9\) \(C\) is the event \(- 1.5 < Z < 1.5\)

1. Find
   1. \(\mathrm { P } ( A )\)
   2. \(\mathrm { P } ( B )\)
   3. \(\mathrm { P } ( C )\)
   4. \(\mathrm { P } ( A \cup C )\) The random variable \(X\) has a normal distribution with mean 21 and standard deviation 5
2. Find the value of \(w\) such that \(\mathrm { P } ( X > w \mid X > 28 ) = 0.625\)

5. A midwife records the weights, in kg , of a sample of 50 babies born at a hospital. Her results are given in the table below. [You may use \(\sum \mathrm { f } x ^ { 2 } = 611.375\) ] A histogram has been drawn to represent these data. The bar representing the weight \(2 \leqslant w < 3\) has a width of 1 cm and a height of 4 cm .

1. Calculate the width and height of the bar representing a weight of \(3 \leqslant w < 3.5\)
2. Use linear interpolation to estimate the median weight of these babies.
3. 1. Show that an estimate of the mean weight of these babies is 3.43 kg .
   2. Find an estimate of the standard deviation of the weights of these babies. Shyam decides to model the weights of babies born at the hospital, by the random variable \(W\), where \(W \sim \mathrm {~N} \left( 3.43,0.65 ^ { 2 } \right)\)
4. Find \(\mathrm { P } ( W < 3 )\)
5. With reference to your answers to (b), (c)(i) and (d) comment on Shyam's decision. A newborn baby weighing 3.43 kg is born at the hospital.
6. Without carrying out any further calculations, state, giving a reason, what effect the addition of this newborn baby to the sample would have on your estimate of the
   1. mean,
   2. standard deviation.

6. The time, in minutes, taken by men to run a marathon is modelled by a normal distribution with mean 240 minutes and standard deviation 40 minutes.

1. Find the proportion of men that take longer than 300 minutes to run a marathon.
   (3) Nathaniel is preparing to run a marathon. He aims to finish in the first 20\% of male runners.
2. Using the above model estimate the longest time that Nathaniel can take to run the marathon and achieve his aim.
   (3) The time, \(W\) minutes, taken by women to run a marathon is modelled by a normal distribution with mean \(\mu\) minutes. Given that \(\mathrm { P } ( W < \mu + 30 ) = 0.82\)
3. find \(\mathrm { P } ( W < \mu - 30 \mid W < \mu )\)

2. An estate agent is studying the cost of office space in London. He takes a random sample of 90 offices and calculates the cost, \(\pounds x\) per square foot. His results are given in the table below. A histogram is drawn for these data and the bar representing \(50 \leqslant x < 60\) is 2 cm wide and 8 cm high.

1. Calculate the width and height of the bar representing \(20 \leqslant x < 40\)
2. Use linear interpolation to estimate the median cost.
3. Estimate the mean cost of office space for these data.
4. Estimate the standard deviation for these data.
5. Describe, giving a reason, the skewness. Rika suggests that the cost of office space in London can be modelled by a normal distribution with mean \(\pounds 50\) and standard deviation \(\pounds 10\)
6. With reference to your answer to part (e), comment on Rika's suggestion.
7. Use Rika's model to estimate the 80th percentile of the cost of office space in London.

5. Yuto works in the quality control department of a large company. The time, \(T\) minutes, it takes Yuto to analyse a sample is normally distributed with mean 18 minutes and standard deviation 5 minutes.

1. Find the probability that Yuto takes longer than 20 minutes to analyse the next sample. (3) The company has a large store of samples analysed by Yuto with the time taken for each analysis recorded. Serena is investigating the samples that took Yuto longer than 15 minutes to analyse. She selects, at random, one of the samples that took Yuto longer than 15 minutes to analyse.
2. Find the probability that this sample took Yuto more than 20 minutes to analyse. Serena can identify, in advance, the samples that Yuto can analyse in under 15 minutes and in future she will assign these to someone else.
3. Estimate the median time taken by Yuto to analyse samples in future.

3. The random variable \(Y\) has a normal distribution with mean \(\mu\) and standard deviation \(\sigma\) The \(\mathrm { P } ( Y > 17 ) = 0.4\) Find

1. \(\mathrm { P } ( \mu < Y < 17 )\)
2. \(\mathrm { P } ( \mu - \sigma < Y < 17 )\)

7. Farmer Adam grows potatoes. The weights of potatoes, in grams, grown by Adam are normally distributed with a mean of 140 g and a standard deviation of 40 g . Adam cannot sell potatoes with a weight of less than 92 g .

1. Find the percentage of potatoes that Adam grows but cannot sell. The upper quartile of the weight of potatoes sold by Adam is \(q _ { 3 }\)
2. Find the probability that the weight of a randomly selected potato grown by Adam is more than \(q _ { 3 }\)
3. Find the lower quartile, \(q _ { 1 }\), of the weight of potatoes sold by Adam. Betty selects a random sample of 3 potatoes sold by Adam.
4. Find the probability that one weighs less than \(q _ { 1 }\), one weighs more than \(q _ { 3 }\) and one has a weight between \(q _ { 1 }\) and \(q _ { 3 }\)

   END

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 \(\mathrm { P } ( 69 \leq X \leq 83 )\).

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 \(p\).

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.
   

3.

1. State the condition under which the normal distribution may be used as an approximation to the Poisson distribution. The number of reported first aid incidents per week at an airport terminal has a Poisson distribution with mean 3.5
2. Find the modal number of reported first aid incidents in a randomly selected week. Justify your answer. The random variable \(X\) represents the number of reported first aid incidents at this airport terminal in the next 2 weeks.
3. Find \(\mathrm { P } ( X > 5 )\)
4. Given that there were exactly 6 reported first aid incidents in a 2 week period, find the probability that exactly 4 were reported in the first week.
5. Using a suitable approximation, find the probability that in the next 40 weeks there will be at least 120 reported first aid incidents.
   

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.