Normal distribution

Exercise 1

The owner of a factory demands that no more than 5% of all the packs of sugar produced should weigh less than the desired 1kg.

a

What is the mean filling weight you have to set the machine to if your standard deviation is $4$ gram (the precision of the filling machine)?

b

European Union regulations are even stricter: no more than 2% of packs is allowed to be too light. What filling weight does the factory now have to set the machine to?

The owner of the factory is not happy about the higher filling weight, since this means higher production costs. After all he has to put on average more sugar into every pack. He therefore decides not to increase the filling weight, but to adjust the precision of the machine. The mean filling weight he wants to use is $1003$ gram and he is taking into account that the EU allows up to 2% of packs to be too light.

c

What is the maximum standard deviation the machine can still cause in order to adhere to the regulations?

Exercise 2

In the production line of a particular type of car, the steering wheel is put into place manually. This step in the production process takes on average $55$ seconds. The time taken $T$ seems to be approximately normally distributed about this mean, with a standard deviation of $4$ seconds.

a

In a given month $1200$ cars of this type are produced. Give an estimate of the number of cars where the placing of the steering wheel took longer than $60$ seconds.

b

How long did the fastest 5% of processes take?

c

The owner of the factory wants to find out if machines could perform better than people. The mean placing time of the machine is still $55$ seconds, but the standard deviation becomes much smaller. Now only 1% of all placings take longer than $60$ seconds. What is the standard deviation of the mechanical placing?

Exercise 3

A factory produces screws of different sizes. There is an order for a delivery of screws with a head diameter between $9.98$ mm and $10.03$ mm. Screws with heads that are thicker or thinner will be rejected. The owner sets the machine to produce an average diameter of $9.99$ mm. The standard deviation of the machine is $0.02$ mm.

a

What percentage of the screws will have an acceptable diameter?

b

What percentage of screws will have an acceptable diameter if the standard deviation of the machine could be reduced to $0.01$ mm?

The producer would like 99% of screws to have an acceptable diameter. He thinks that he could achieve this by adjusting the mean diameter of the machine. He also wants to improve the adjustment of the machine so that the standard deviation is reduced.

c

What setting for the mean and what standard deviation will he need in order to achieve the 99% target?

Exercise 4

At a seed improvement company, the lengths of a particular type of plant are measured. The lengths of the plants appear to be normally distributed. 12.5% of the plants is longer than $60$ cm and 39% of the plants is shorter than $30$ cm.

a

Calculate the mean and standard deviation of the length of this type of plant.

Plants that are too small cannot be used for seed production and are destroyed. It turns out that 30% of plants has to be destroyed.

b

Up to what length are plants considered too small?

Exercise 5

A baker produces Christmas cakes of $1000$ g each.

a

What is the standard deviation if the mean weight of the cakes is $1000$ g and 5% of the loaves weighs less than $900$ g?

b

If the standard deviation is $60$ g , what percentage of cakes weighs less than $900$ g?

c

What is the average weight of a cake if the standard deviation is $65$ g and 5% of the cakes weigh less than $900$ g?

Exercises