The normal distribution and when to use it Problem set A

A Galton board has 10 rows of pins. At each pin a ball goes left or right with probability 0.5, independently. Let B be the number of times a ball goes right, so \(B \sim B(10, 0.5)\).

1. (a) Find the mean and standard deviation of B.
2. (b) Many balls are dropped. Describe the shape of the pile of balls and explain why it has that shape.

The length of a salmon is normally distributed with mean 60 cm and standard deviation 4 cm. Use the 68–95–99.7 rule.

1. (a) Find the percentage between 52 and 68 cm.
2. (b) Find the percentage below 56 cm.
3. (c) Find the percentage between 60 and 68 cm.

The box plot shows the time in seconds that 80 web pages take to load. Is a normal distribution a suitable model? Give reasons.

The box plot shows the masses in kg of 100 sheep. Is a normal distribution a suitable model? Give a reason.

The time, in minutes, that a customer waits in a queue is modelled by a normal distribution with mean 4.5 and standard deviation 2.8.

1. (a) How many standard deviations is the limit 0 below the mean?
2. (b) Explain why a normal distribution is not a good model. Use your answer to part (a).

The mass of a jar of jam is normally distributed with mean 250 g and standard deviation 5 g.

1. (a) Use the 68–95–99.7 rule to give an interval that contains about 99.7% of the values.
2. (b) One jar measures 268 g. How many standard deviations from the mean is that? Is it unusual?

A Galton board is tilted so that a ball goes right with probability 0.2 at each of its 10 rows of pins. Let B be the number of times a ball goes right, so \(B \sim B(10, 0.2)\).

1. (a) Find the mean and standard deviation of B.
2. (b) How many standard deviations is 0 below the mean?
3. (c) Explain why the pile of balls is not a normal shape.