IB Maths AI HL – Topic 4: Statistics and Probability

The number of emails, $X$, received by an office in a one-hour period follows a Poisson distribution with mean $3.2$.(a) Find $P(X = 5)$.(b) Find $P(X \leq 2)$.(c) State $\text{Var}(X)$.

Calls to a helpline arrive independently at a mean rate of $2.1$ calls per $10$-minute period, and can be modelled by a Poisson distribution.Find the probability that at least one call is received in a randomly chosen $10$-minute period.

Let $X \sim \text{Po}(3.2)$ and $Y \sim \text{Po}(2.1)$ be independent random variables, representing the number of emails and calls received in an hour respectively. Let $T = X + Y$.(a) State the distribution of $T$, including its parameter.(b) Find $P(T = 4)$.

The number of flaws, $W$, in a metre of fabric follows a Poisson distribution with mean $m$. Given that $P(W = 0) = 0.0498$, find the value of $m$.

A random sample of $40$ students sat a test. The sample mean score was $68$ and the sample standard deviation was $12$.Assuming the sample size is large enough to use a normal approximation, construct a $95\%$ confidence interval for the population mean score, giving the endpoints correct to one decimal place.

A random sample of $15$ observations from a normal population has mean $52$ and standard deviation $6$. It is claimed the population mean is $48$. Test, at the $5\%$ significance level, whether the sample provides evidence that the population mean exceeds $48$.State your hypotheses, calculate the test statistic, and state your conclusion (critical value $t_{14} = 1.761$ at the $5\%$ level, one-tailed).

Explain what is meant by:(a) a Type I error,(b) a Type II error,in the context of a hypothesis test.

A fair six-sided die is rolled $60$ times, so the expected frequency for each face is $10$. The observed frequencies for faces $1$ to $6$ are $4, 12, 9, 15, 11, 9$.Carry out a $\chi^2$ goodness-of-fit test at the $5\%$ significance level to determine whether the die is fair. The critical value is $\chi^2_5 = 11.07$.

A researcher wants to estimate a population mean, given that the population standard deviation is known to be $8$. Find the minimum sample size required so that a $95\%$ confidence interval for the mean has a margin of error of at most $2$.

In a simplified weather model, if a day is sunny the probability the next day is sunny is $0.7$ (otherwise rainy); if a day is rainy the probability the next day is sunny is $0.4$ (otherwise rainy). Today is sunny.(a) Write down the transition matrix $T$ for this Markov chain, using the state vector $\begin{pmatrix} P(\text{sunny}) \\ P(\text{rainy}) \end{pmatrix}$.(b) Find the probability that it is sunny two days from today.

Using the weather model above, find the long-run (steady-state) probability that a day is sunny.

Explain, in the context of the weather model, what is meant by a steady-state (equilibrium) probability vector, and state one condition on the transition matrix needed for a unique steady-state to exist.