Normal distribution

The random variable X has the distribution N(45, 4²).

Find P(X > 50).

The random variable Y has the distribution N(25, σ²).

Given that P(Y < 30) = 0.75, find σ.

The masses of stones on a certain beach are normally distributed with mean 0.36 kg and standard deviation 0.12 kg.

Find the probability that two stones chosen at random from this beach both have masses

between 0.3 kg and 0.4 kg.

Determine the probability that the mean mass of a random sample of 40 stones from this

beach is less than 0.32 kg.

The random variable Y has the distribution N(μ, σ²).

Find P(Y > μ - σ).

Given that P(Y > 45) = 0.2 and P(Y < 2 5) = 0. 3, determine the values of μ and σ.

The random variables U and V have the distributions N(10, 4) and N(12, 9) respectively.

It is given that P(U < b) = P(V > c), where b > 10 and c < 12.

Determine b in terms of c.

The heights, in centimetres, of a random sample of 150 plants of a certain variety were measured. The results are summarised in the histogram.

Frequency

One of the 150 plants is chosen at random, and its height, X cm, is noted.

Show that P(20 < X < 30) = 0.147, correct to 3 significant figures.

Sam suggests that the distribution of X can be well modelled by the distribution N(40, 100).

Give a brief justification for the use of the normal distribution in this context.

Give a brief justification for the choice of the parameter values 40 and 100.

Use Sam’s model to find P (20 < X < 30).

Nina suggests a different model. She uses the midpoints of the classes to calculate estimates, m and s, for the mean and standard deviation respectively, in centimetres, of the 150 heights. She then uses the distribution N(m, s²) as her model.

Use Nina’s model to find P (20 < X < 30).

Complete the table in the Printed Answer Booklet to show the probabilities obtained

from Sam’s model and Nina’s model.

By considering the different ranges of values of X given in the table, discuss how well

the two models fit the original distribution.

As part of a research project, the masses, m grams, of a random sample of 1000 pebbles from a certain beach were recorded. The results are summarised in the table.

Calculate estimates of the mean and standard deviation of these masses.

The masses, x grams, of a random sample of 1000 pebbles on a different beach were also found. It was proposed that the distribution of these masses should be modelled by the random variable X ∼ N(200, 3600).

Use the model to find P (150 < X < 210).

Use the model to determine x₁ such that P(160 < X < x₁) = 0.6, giving your answer correct

to five significant figures.

It was found that the smallest and largest masses of the pebbles in this second sample were 112 g and 288 g respectively.

Use these results to show that the model may not be appropriate.

Suggest a different value of a parameter of the model in the light of these results.

The variable X has the distribution N(20, 9).

Find P(X > 25).

Given that P(X > a) = 0.2, find a.

Find b such that P(20 - b < X < 20 + b) = 0.5.