Inverse normal and normal practice Problem set A

The height of a variety of sunflower is normally distributed with mean 210 cm and standard deviation 18 cm. The shortest 15% of plants are given extra fertiliser. Find the height below which a plant is given extra fertiliser.

Scores on an aptitude test are normally distributed with mean 100 and standard deviation 15. The top 8% of scores are invited to an interview. Find the lowest score that gets an interview.

\(X \sim N(120, 15^2)\). Find the value of k such that \(P(110 < X < k) = 0.35\).

The time taken by athletes to complete a 5 km run is normally distributed with mean 26.0 minutes and standard deviation 3.2 minutes.

1. (a) Find the lower quartile.
2. (b) Find the upper quartile.
3. (c) Find the interquartile range.

The daily sales of a café are normally distributed with mean 180 cups and standard deviation 25 cups. Find the probability that the café sells more than 200 cups on a day.

The mass of a strawberry is normally distributed with mean 24 g and standard deviation 3.5 g. In a delivery of 800 strawberries, find the expected number heavier than 30 g.

The time, in seconds, that a swimmer takes to swim 50 m is normally distributed with mean 31.4 and standard deviation 1.2. A coach calls the middle 60% of her times typical. Find the shortest and longest typical time.

\(X \sim N(\mu, 6^2)\) and \(P(X < 40) = 0.3\). Find the value of \(\mu\).

The time a player takes to complete a puzzle is normally distributed with mean 38 minutes and standard deviation 7 minutes. 90% of players finish within t minutes. Find t.

The mass of a tomato is normally distributed with mean 70 g and standard deviation 12 g. A grower sorts 2 000 tomatoes. Small tomatoes (under 55 g) sell for €0.20 each, medium tomatoes (55 g to 80 g) for €0.35 each and large tomatoes (over 80 g) for €0.50 each. Find the expected income.

The time, in minutes, that a customer waits for a call to be answered is modelled by a normal distribution with mean 6 and standard deviation 3.5.

1. (a) Find the probability that the model gives a negative value.
2. (b) What does your answer tell you about the model?

The diameter of a drill bit is normally distributed with mean 8.00 mm and standard deviation 0.05 mm. A drill bit is defective if its diameter is less than 7.9 mm. A batch of 10 bits is taken.

1. (a) Find the probability that one is below the limit given in the question.
2. (b) Find the probability that at least one bit in the batch is defective.
3. (c) State two assumptions you made in part (b).

A machine pours coffee into cups. The volume is normally distributed with mean 180 ml. 3% of cups contain less than 170 ml.

1. (a) Find the standard deviation.
2. (b) Find the probability that a cup holds more than 188 ml.

The masses of 400 rams are normally distributed with mean 75 kg and standard deviation 6 kg. The lightest ram has mass 60 kg and the heaviest 91 kg. Complete a box plot of the masses.

1. (a) Find the median, lower quartile and upper quartile.
2. (b) Write the five numbers for the box plot and describe its shape.

The time in minutes that a customer spends in a shop is modelled by a normal distribution with mean 5.2 and standard deviation 3.1.

1. (a) How many standard deviations is the mean above 0?
2. (b) Explain why a normal distribution is not a suitable model.

The height, H metres, of the trees in a plantation is normally distributed. 12% of trees are shorter than 9 m and 30% are taller than 14 m.

1. (a) Find the mean and the standard deviation.
2. (b) Find the probability that a tree is more than 16 m.