A-Level Further Maths: Further Statistics Practice

A discrete random variable X has PMF P(X=x) = kx for x = 1, 2, 3. Find the value of k.

Given E(X) = 3 and Var(X) = 5, find E(X²).

If Y = 3X - 2 where E(X) = 4 and Var(X) = 9, find E(Y) and Var(Y).

X and Y are independent discrete random variables with Var(X) = 4 and Var(Y) = 9. Find Var(2X - 3Y).

A continuous random variable X has PDF f(x) = kx for 0 ≤ x ≤ 2. Find the value of k.

X follows a continuous uniform distribution U(a, b). What is Var(X)?

The lifetime of a component (in hours) follows an exponential distribution with mean 200 hours. What is the probability it lasts more than 250 hours?

X has PDF f(x) = 3x² for 0 ≤ x ≤ 1. Find the median of X.

When should you use a t-test instead of a z-test for testing a population mean?

In a chi-squared goodness of fit test with 5 categories, the observed frequencies are 20, 30, 25, 15, 10 and the expected frequencies are 20, 20, 20, 20, 20. Calculate the chi-squared test statistic.

A contingency table has 3 rows and 4 columns. How many degrees of freedom does the chi-squared test for association have?

In a chi-squared goodness of fit test, what is the null hypothesis?

Define Type I error in the context of hypothesis testing.

A test has significance level α = 0.05. If the power of the test is 0.8, what is the probability of a Type II error?

For a given significance level α, which change would increase the power of a hypothesis test?

The operating characteristic (OC) curve plots which two quantities?