Newcastle University Mathematics and Statistics

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.

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

A multiple linear regression model of the form:

\[Y=\beta_0+\beta_1X_1+ \beta_2X_2+\beta_3X_3+\beta_4X_4+\epsilon \]

is fitted to some data in Minitab which generates a table showing estimates of the parameters with associated $p$-values. Determine which variable to exclude first.

One-way ANOVA example

Two factor ANOVA example

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

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.

A graphical approach to aiding students in writing down a formal proof of discontinuity of a function at a given point.

Uses JSXgraph to sketch the graphs and involves some interaction/experimentation by students in finding appropriate intervals.

No description given

Solve a system of linear equations using Gaussian elimination.

For a sample of size n from a normal distribution and given the mean of the sample and the standard deviation of the distribution, find the MLE for the mean. Also the expected information and a confidence interval for the mean.

For a sample of size n from a Poisson distribution $\operatorname{Poisson}(\lambda)$ and given the mean of the sample, find the MLE for $\lambda$. Also find the expected information and a confidence interval for $\lambda$.

For a sample of size n from an Exponential distribution $\operatorname{Exponential}(\lambda)$  and given the mean of the sample, find the MLE for $\lambda$. Also find the expected information and a confidence interval for $\lambda$.

For a sample of size n from a normal distribution, given mean of the sample mean and the standard deviation , find the t-statistic corresponding to a null hypothesis $\mu=m$ and a given confidence level. Check if the result is significant at this level.

Given a linear combination $Y$ of three independent random variables with given means and variances, find the mean of variance of $Y$.

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

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

Looking up t-tables. 

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?

Entering numbers in Numbas, Part 1.

Details on inputting numbers into Numbas.

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 $t$. 

Looking up t-tables. 

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.

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

Enumerate the elements in some sets defined using set builder notation.