Binomial 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.

A school contains 500 students in years 7 to 11 and 250 students in years 12 and 13. A random sample of 20 students is selected to represent the school at a parents’ evening. The number of students in the sample who are from years 12 and 13 is denoted by X.

State a suitable binomial model for X.

Use your model to answer the following.

Write down an expression for P(X = x).

State, in set notation, the values of x for which your expression is valid.

Find P(5 ⩽ X ⩽ 9).

State one disadvantage of using a random sample in this context.

A firm claims that no more than 2% of their packets of sugar are underweight. A market researcher believes that the actual proportion is greater than 2%. In order to test the firm’s claim, the researcher weighs a random sample of 600 packets and carries out a hypothesis test, at the 5% significance level, using the null hypothesis p = 0.02.

Given that the researcher’s null hypothesis is correct, determine the probability that the

researcher will conclude that the firm’s claim is incorrect.

The researcher finds that 18 out of the 600 packets are underweight. A colleague says

“18 out of 600 is 3%, so there is evidence that the actual proportion of underweight bags is greater than 2%.”

Criticise this statement.

The values of three independent observations of X are denoted by X₁, X₂ and X₃.

Given that X₁ + X₂ + X₃ = 9, determine the probability that at least one of these three values is

equal to 2.

Freda chooses values of X at random until she has obtained X = 2 exactly three times. She then stops.

Determine the probability that she chooses exactly 10 values of X.

In this question you must show detailed reasoning.

The probability that Paul’s train to work is late on any day is 0.15, independently of other days.

The number of days on which Paul’s train to work is late during a 450-day period is denoted by the random variable Y. Find a value of a such that P(Y > a) ≈

The number of days on which Paul’s train to work is late during a 50-day period is modelled by the random variable X.

Find the values of r for which P (X = r) ⩽ P (X = r + 1).

Hence find the most likely number of days on which the train will be late during a 50-day period.