Inverse normal and normal practice Problem set B

The time a runner takes to complete a 200 m race is normally distributed with mean 25.4 s and standard deviation 1.2 s. The fastest 5% of runners reach the final. Find the slowest time that reaches the final.

The monthly pay of workers in a factory is normally distributed with mean €2 400 and standard deviation €350. The top 12% of earners pay a higher rate of tax. Find the lowest pay that is taxed at the higher rate.

\(X \sim N(45, 6^2)\). Find the value of k such that \(P(40 < X < k) = 0.45\).

The lengths of rope cut by a machine are normally distributed with mean 2.40 m and standard deviation 0.06 m.

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

The lifetime of a smartphone battery is normally distributed with mean 820 days and standard deviation 90 days. Find the probability that a battery lasts less than 700 days.

The length of a pencil after sharpening is normally distributed with mean 17.5 cm and standard deviation 0.8 cm. In a box of 1 200 pencils, find the expected number longer than 18.5 cm.

The braking distance of a model of car from 50 km/h is normally distributed with mean 24.5 m and standard deviation 1.8 m. The middle 80% of distances are called usual. Find the shortest and longest usual distance.

\(X \sim N(\mu, 4.5^2)\) and \(P(X < 62) = 0.15\). Find the value of \(\mu\).

The time a student takes to complete a cross-country course is normally distributed with mean 52 minutes and standard deviation 6 minutes. 90% of students finish within t minutes. Find t.

The mass of a peach is normally distributed with mean 150 g and standard deviation 20 g. A grower sorts 3 000 peaches. Small peaches (under 130 g) sell for €0.25 each, medium peaches (130 g to 170 g) for €0.40 each and large peaches (over 170 g) for €0.60 each. Find the expected income.

The distance, in kilometres, that a delivery driver travels to a customer is modelled by a normal distribution with mean 4 and standard deviation 2.2.

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

The mass of a bag from a packing machine is normally distributed with mean 1 015 g and standard deviation 12 g. A bag is underweight if its mass is less than 1 000 g. A sample of 15 bags 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 2 bags in the sample are underweight.
3. (c) State two assumptions you made in part (b).

A machine fills bottles with juice. The volume is normally distributed with mean 330 ml. 2% of bottles contain less than 325 ml.

1. (a) Find the standard deviation.
2. (b) Find the probability that a bottle holds more than 333 ml.

The lengths of 500 trout are normally distributed with mean 32 cm and standard deviation 4 cm. The shortest trout is 19 cm and the longest 45 cm. Complete a box plot of the lengths.

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 number of days that a parcel is delayed is modelled by a normal distribution with mean 4.0 and standard deviation 2.8.

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

The life, L hours, of a type of battery is normally distributed. 15% of batteries last less than 18 hours and 25% last more than 30 hours.

1. (a) Find the mean and the standard deviation.
2. (b) Find the probability that a battery lasts more than 36 hours.