2.04f Find normal probabilities: Z transformation

Bottles of wine are stacked in racks of 12. The weights of these bottles are normally distributed with mean 1.3 kg and standard deviation 0.06 kg. The weights of the empty racks are normally distributed with mean 2 kg and standard deviation 0.3 kg.

1. Find the probability that the total weight of a full rack of 12 bottles of wine is between 17 kg and 18 kg. [5]
2. Two bottles of wine are chosen at random. Find the probability that they differ in weight by more than 0.05 kg. [5]

A machine squeezes apples to extract their juice. The volume of juice, \(J\) ml, extracted from 1 kg of apples is modelled by a normal distribution with mean \(\mu\) and standard deviation \(\sigma\) Given that \(\mu = 500\) and \(\sigma = 25\) use standardisation to

1. 1. show that P\((J > 510) = 0.3446\) [2]
   2. calculate the value of \(d\) such that P\((J > d) = 0.9192\) [3]

Zen randomly selects 5 bags each containing 1 kg of apples and records the volume of juice extracted from each bag of apples.

1. Calculate the probability that each of the 5 bags of apples produce less than 510ml of juice. [2]

Following adjustments to the machine, the volume of juice, \(R\) ml, extracted from 1 kg of apples is such that \(\mu = 520\) and \(\sigma = k\) Given that P\((R < r) = 0.15\) and P\((R > 3r - 800) = 0.005\)

1. find the value of \(r\) and the value of \(k\) [7]

The duration of the pregnancy of a certain breed of cow is normally distributed with mean \(\mu\) days and standard deviation \(\sigma\) days. Only 2.5\% of all pregnancies are shorter than 235 days and 15\% are longer than 286 days.

1. Show that \(\mu - 235 = 1.96\sigma\). [2]
2. Obtain a second equation in \(\mu\) and \(\sigma\). [3]
3. Find the value of \(\mu\) and the value of \(\sigma\). [4]
4. Find the values between which the middle 68.3\% of pregnancies lie. [2]

The heights of a population of women are normally distributed with mean \(\mu\) cm and standard deviation \(\sigma\) cm. It is known that 30% of the women are taller than 172 cm and 5% are shorter than 154 cm.

1. Sketch a diagram to show the distribution of heights represented by this information. [3]
2. Show that \(\mu = 154 + 1.6449\sigma\). [3]
3. Obtain a second equation and hence find the value of \(\mu\) and the value of \(\sigma\). [4]

A woman is chosen at random from the population.

1. Find the probability that she is taller than 160 cm. [3]

The random variable \(X \sim \text{N}(\mu, 5^2)\) and \(\text{P}(X < 23) = 0.9192\)

1. Find the value of \(\mu\). [4]
2. Write down the value of \(\text{P}(\mu < X < 23)\). [1]

Past records show that the times, in seconds, taken to run 100 m by children at a school can be modelled by a normal distribution with a mean of 16.12 and a standard deviation of 1.60 A child from the school is selected at random.

1. Find the probability that this child runs 100 m in less than 15 s. [3]

On sports day the school awards certificates to the fastest 30\% of the children in the 100 m race.

1. Estimate, to 2 decimal places, the slowest time taken to run 100 m for which a child will be awarded a certificate. [4]

Strips of metal are cut to length \(L\) cm, where \(L \sim N(\mu, 0.5^2)\).

1. Given that 2.5\% of the cut lengths exceed 50.98 cm, show that \(\mu = 50\). [5]
2. Find \(P(49.25 < L < 50.75)\). [4]

Those strips with length either less than 49.25 cm or greater than 50.75 cm cannot be used. Two strips of metal are selected at random.

1. Find the probability that both strips cannot be used. [2]

A group of students believes that the time taken to travel to college, \(T\) minutes, can be assumed to be normally distributed. Within the college 5\% of students take at least 55 minutes to travel to college and 0.1\% take less than 10 minutes. Find the mean and standard deviation of \(T\). [9]

A garden centre sells canes of nominal length 150 cm. The canes are bought from a supplier who uses a machine to cut canes of length L where L ~ N(\(\mu\), 0.3²).

1. Find the value of \(\mu\), to the nearest 0.1 cm, such that there is only a 5\% chance that a cane supplied to the garden centre will have length less than 150 cm. [4]

A customer buys 10 of these canes from the garden centre.

1. Find the probability that at most 2 of the canes have length less than 150 cm. [3]

Another customer buys 500 canes.

1. Using a suitable approximation, find the probability that fewer than 35 of the canes will have length less than 150 cm. [6]

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, [3]
2. more than three. [2]

A customer bought three boxes.

1. Find the probability that only 2 of the boxes contained exactly 1 egg with a double yolk. [3]

The farmer delivered 10 boxes to a local shop.

1. Using a suitable approximation, find the probability that the delivery contained at least 9 eggs with double yolks. [4]

The weight of an individual egg can be modelled by a normal distribution with mean 65 g and standard deviation 2.4 g.

1. Find the probability that a randomly chosen egg weighs more than 68 g. [3]

1. Write down the condition needed to approximate a Poisson distribution by a Normal distribution. [1]

The random variable Y ~ Po(30).

1. Estimate P(Y > 28). [6]

The random variable R has the binomial distribution B(12, 0.35).

1. Find P(R ≥ 4). [2]

The random variable S has the Poisson distribution with mean 2.71.

1. Find P(S ≤ 1). [3]

The random variable T has the normal distribution N(2.5, 5²).

1. Find P(T ≤ 18). [2]

The random variable \(R\) has the binomial distribution B(12, 0.35).

1. Find P(\(R \geq 4\)). [2]

The random variable \(S\) has the Poisson distribution with mean 2.71.

1. Find P(\(S \leq 1\)). [3]

The random variable \(T\) has the normal distribution N(25, \(5^2\)).

1. Find P(\(T \leq 18\)). [2]

The discrete random variable \(X\) is distributed B(\(n\), \(p\)).

1. Write down the value of \(p\) that will give the most accurate estimate when approximating the binomial distribution by a normal distribution. [1]
2. Give a reason to support your value. [1]
3. Given that \(n = 200\) and \(p = 0.48\), find P(\(90 \leq X < 105\)). [7]

A web server is visited on weekdays, at a rate of 7 visits per minute. In a random one minute on a Saturday the web server is visited 10 times.

1. 1. Test, at the 10\% level of significance, whether or not there is evidence that the rate of visits is greater on a Saturday than on weekdays. State your hypotheses clearly.
   2. State the minimum number of visits required to obtain a significant result.
   [7]
2. State an assumption that has been made about the visits to the server. [1]

In a random two minute period on a Saturday the web server is visited 20 times.

1. Using a suitable approximation, test at the 10\% level of significance, whether or not the rate of visits is greater on a Saturday. [6]

A disease occurs in 3\% of a population.

1. State any assumptions that are required to model the number of people with the disease in a random sample of size \(n\) as a binomial distribution. [2]
2. Using this model, find the probability of exactly 2 people having the disease in a random sample of 10 people. [3]
3. Find the mean and variance of the number of people with the disease in a random sample of 100 people. [2]

A doctor tests a random sample of 100 patients for the disease. He decides to offer all patients a vaccination to protect them from the disease if more than 5 of the sample have the disease.

1. Using a suitable approximation, find the probability that the doctor will offer all patients a vaccination. [3]

A garden centre sells canes of nominal length 150 cm. The canes are bought from a supplier who uses a machine to cut canes of length \(L\) where \(L \sim \mathrm{N}(\mu, 0.3^2)\).

1. Find the value of \(\mu\), to the nearest 0.1 cm, such that there is only a 5\% chance that a cane supplied to the garden centre will have length less than 150 cm. [4]

A customer buys 10 of these canes from the garden centre.

1. Find the probability that at most 2 of the canes have length less than 150 cm. [3]

Another customer buys 500 canes.

1. Using a suitable approximation, find the probability that fewer than 35 of the canes will have length less than 150 cm. [6]

In Manuel's restaurant the probability of a customer asking for a vegetarian meal is 0.30. During one particular day in a random sample of 20 customers at the restaurant 3 ordered a vegetarian meal.

1. Stating your hypotheses clearly, test, at the 5\% level of significance, whether or not the proportion of vegetarian meals ordered that day is unusually low. [5]

Manuel's chef believes that the probability of a customer ordering a vegetarian meal is 0.10. The chef proposes to take a random sample of 100 customers to test whether or not there is evidence that the proportion of vegetarian meals ordered is different from 0.10.

1. Stating your hypotheses clearly, use a suitable approximation to find the critical region for this test. The probability for each tail of the region should be as close as possible to 2.5\%. [6]
2. State the significance level of this test giving your answer to 2 significant figures. [1]

A biologist is studying the behaviour of sheep in a large field. The field is divided up into a number of equally sized squares and the average number of sheep per square is 2.25. The sheep are randomly spread throughout the field.

1. Suggest a suitable model for the number of sheep in a square and give a value for any parameter or parameters required. [1]

Calculate the probability that a randomly selected sample square contains

1. no sheep, [1]
2. more than 2 sheep. [4]

A sheepdog has been sent into the field to round up the sheep.

1. Explain why the model may no longer be applicable. [1]

In another field, the average number of sheep per square is 20 and the sheep are randomly scattered throughout the field.

1. Using a suitable approximation, find the probability that a randomly selected square contains fewer than 15 sheep. [7]

A farm produces potatoes. The potatoes are packed into sacks. The weight of a sack of potatoes is modelled by a normal distribution with mean 25.6 kg and standard deviation 0.24 kg

1. Find the probability that two randomly chosen sacks of potatoes differ in weight by more than 0.5 kg [6]

Sacks of potatoes are randomly selected and packed onto pallets. The weight of an empty pallet is modelled by a normal distribution with mean 20.0 kg and standard deviation 0.32 kg Each full pallet of potatoes holds 30 sacks of potatoes.

1. Find the probability that the total weight of a randomly chosen full pallet of potatoes is greater than 785 kg [5]

A fair six-sided die is labelled with the numbers 1, 2, 3, 4, 5 and 6. The die is rolled 40 times and the score, \(S\), for each roll is recorded.

1. Find the mean and the variance of \(S\). [2]
2. Find an approximation for the probability that the mean of the 40 scores is less than 3 [3]

The weights of a group of males are normally distributed with mean 80 kg and standard deviation 2.6 kg. A random sample of 10 of these males is selected.

1. Write down the distribution of \(M\), the mean weight, in kg, of this sample. [2]
2. Find P(\(M < 78.5\)). [3]

The weights of a group of females are normally distributed with mean 59 kg and standard deviation 1.9 kg. A random sample of 6 of the males and 4 of the females enters a lift that can carry a maximum load of 730 kg.

1. Find the probability that the maximum load will be exceeded when these 10 people enter the lift. [5]

The weights of tubs of margarine are known to be normally distributed. A random sample of 10 tubs of margarine were weighed, to the nearest gram, and the results were as follows. $$498 \quad 502 \quad 500 \quad 496 \quad 509 \quad 504 \quad 511 \quad 497 \quad 506 \quad 499$$

1. Find unbiased estimates of the mean and the variance of the population from which this sample was taken. [5]

Given that the population standard deviation is 5.0 g,

1. estimate limits, to 2 decimal places, between which 90\% of the weights of the tubs lie, [2]
2. find a 95\% confidence interval for the mean weight of the tubs. [5]

A second random sample of 15 tubs was found to have a mean weight of 501.9 g.

1. Stating your hypotheses clearly and using a 1\% level of significance, test whether or not the mean weight of these tubs is greater than 500 g. [5]

The weights of a group of males are normally distributed with mean 80 kg and standard deviation 2.6 kg. A random sample of 10 of these males is selected.

1. Write down the distribution of \(\bar{M}\), the mean weight, in kg, of this sample. [2]
2. Find P(\(\bar{M} < 78.5\)). [3]

The weights of a group of females are normally distributed with mean 59 kg and standard deviation 1.9 kg. A random sample of 6 of the males and 4 of the females enters a lift that can carry a maximum load of 730 kg.

1. Find the probability that the maximum load will be exceeded when these 10 people enter the lift. [5]