$X \sim \operatorname{Binomial}(n,p)$. Find $P(X=a)$, $P(X \leq b)$, $E[X],\;\operatorname{Var}(X)$.

rebelmaths

Two factor ANOVA example

Two-way ANOVA example, 5 subjects, 4 treatments.

Given mean and sd of 1000 sample returns on a scale of 1 to 7 together with a given score, find the z-score.

Also find the 95% confidence interval for the population mean.

Given data on population mean and population standard deviation and three sampling sizes, calculate the probabilities that the sample means are within a specified distance from the population mean.

One-way ANOVA example

Find a regression equation given 12 months data on temperature and sales of a drink. 

Asking users to interpret a minitab output to give the coefficients of a multiple regression together with a prediction based on the subsequent equation.

Minitab was used to fit both an AR(1) model and an AR(2) to a stationary series. A  table is given summarising the results obtained from Minitab. Choose the most appropriate model and make a forecast based on that model.

Minitab was used to fit an AR(1) model to a stationary time series. Given the output answer the following questions about the model and use the model to make forecasts.

Paired t-test to see if there is a difference between times take in a task.

LSD and Tukey yardsticks on three treatments. Also one-way Anova test on same set of data.

LSD and Tukey yardsticks on five treatments. Also two-way Anova test on same set of data.

Given sum of sample from a Normal distribution with unknown mean $\mu$ and known variance $\sigma^2$. Find MLE of $\mu$ and one of four functions of $\mu$.

Using a random sample from a population with given mean and variance, find the expectation and variance of three estimators of $\mu$. Unbiased, efficient?

Given a PDF $f(x)$ on the real line with unknown parameter $t$ and three random observations, find log-likelihood and MLE $\hat{t}$ for $t$. 

Given a discrete random variable $X$ find the expectation of $1/X$ and $e^X$.

Arrivals given by exponential distribution, parameter $\theta$ and $Y$, sample mean on inter-arrival times. Find and calculate unbiased estimator for $\theta$.

Given 3 observations from a $\operatorname{Poisson}(\mu)$ distribution find the likelihood, the log likelihood and the MLE for $\mu$.

Looking up t-tables. 

Looking up t-tables. 

Find a regression equation.

Find a regression equation.

Given sample data find mean, standard deviation, median, interquartile range,

One-way ANOVA example

Sample of size $24$ is given in a table. Find sample mean, sample standard deviation, sample median and the interquartile range.

Two ordered data sets, each with 10 numbers. Find the sample standard deviation for each and for their sum.

Two numbers are drawn at random without replacement from the numbers m to n.

Find the probability that both are odd given their sum is even.

The random variable $X$ has a PDF which involves a parameter $c$. Find the value of $c$. Find the distribution function $F_X(x)$ and $P(a \lt X \lt b)$.

Also find the expectation $\displaystyle \operatorname{E}[X]=\int_{-\infty}^{\infty}xf_X(x)\;dx$.

Provided with information on a sample with sample mean and standard deviation, but no information on the population variance, use the t test to either accept or reject a given null hypothesis.