Normal distribution

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1234Normal probabilities

Exercises

Exercise 1

A machine fills packs of rice of 1kg each. The filling weight that the machine has been set to corresponds to the mean weight of the packs. The weights are normally distributed. The average weight of a pack of rice is $1010$ grams and the standard deviation is $9$ grams.

What percentage of packs weighs less than $1000$ grams?

What percentage of packs weighs more than $1000$ grams?

What is the chance that a pack is too heavy by more than $5$ grams?

What percentage of packs is too heavy by more than $20$ grams?

Exercise 2

There are $20000$ pupils sitting a final maths exam. Their results are approximately normally distributed. The average grade is $6.4$ and the standard deviation is $1.1$

How many candidates have a grade of lower than 5.5 and therefore failed the exam?

How many candidates have a grade of 7.0 or higher?

How many candidates score a 4.0 or lower?

Exercise 3

The filling volume $V$ of a pack of milk is normally distributed with a mean of $1.02$ litres and a standard deviation of $0.015$ litres. The consumer expects that a pack of milk contains $1$ litre.

What percentage of packs contains less than dan $1$ litre of milk?

What percentage of packs contains more than $1.03$ litre of milk?

You just bought such a pack of milk. What is the chance that your pack of milk is $2$ centilitres short?

You cannot determine what percentage of packs contains exactly $1$ litre of milk. You can, however, calculate what percentage of packs contains a volume of $1$ litre rounded to two decimals. That means you are looking at the area between $0.995$ litres and $1.005$ litres. This area has a corresponding percentage. Calculate that percentage.

5% of the packs of milk contains less than a given volume $g$ . Calculate $g$ .

At least how many litres of milk will you find in a pack that belongs to the fullest 10% of packs?

Exercise 4

Research done by Freudenthal en Sittig in 1947 has shown that the heights of women shopping at 'De Bijenkorf' were normally distributed with a mean of $162$ cm and a standard deviation of $6.5$ cm. In answering the questions below, use this normal distribution as a model for the heights of these $5001$ women.

What percentage of women was taller than $170$ cm?

What percentage of women had a height of between $160$ and $170$ cm?

What was your chance to meet a woman in the Bijenkorf who was $160$ cm tall? (Assume all lengths were rounded to whole centimeters.)

How tall were the shortest 10% of women?

How tall were the tallest 10% of women?

Exercise 5

For this question, use the following link:gegevens to open the file "Enkele lichaamsafmetingen van $5001$ vrouwen uit 1947". This file contains a table of knee heights (in cm) of the $5001$ women who participated in the 1947 study by Freudenthal and Sittig, commissioned by De Bijenkorf.

Use a computer to calculate the average knee height and the corresponding standard deviation.

Plot a histogram and approximate this with a normal curve. Indicate both values calculated in a) in this graph.

Now assume that the knee height of women $K$ is normally distributed with your calculated values for the mean $μ$ and the standard deviation $σ$ .

90% of the knee heights lie between $μ - a$ and $μ + a$ . How big is $a$ ?

What is the minimal length of the longest 20% of knee heights?

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