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Interpret Minitab output of multiple regression
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Asking users to interpret a minitab output to give the coefficients of a multiple regression together with a prediction based on the subsequent equation.
Multiple linear regression - decide which variable to exclude
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

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.
Probability sample mean is within d of the population mean
Doesn't work
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

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.
Prove discontinuity of a function at a given point
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

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.
20122013 CBA0_2
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Given a linear combination $Y$ of three independent random variables with given means and variances, find the mean of variance of $Y$ .
Find maximum likelihood estimator of mean of normal distribution
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

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 $\mu$ .
Find the expectation and variance of three estimators of the mean
Has some problems
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Using a random sample from a population with given mean and variance, find the expectation and variance of three estimators of $\mu$ . Unbiased, efficient?
Log-likelihood and maximum likelihood estimator for PDF
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

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 $t$ .
Simplex method
Draft
Exam (2 questions) in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Use the simplex method to solve a linear program.
Approximate the integral of a quadratic by Riemann sums,
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Approximating integral of a quadratic by Riemann sums . Includes an interactive graph in Advice showing the approximations given by the upper and lower sums and how they vary as we increase the number of intervals.
Calculate probabilities from frequency table
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Simple probability question. Counting number of occurences of an event in a sample space with given size and finding the probability of the event.
Find mean, standard deviation, median and interquartile range of sample,
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Given sample data find mean, standard deviation, median, interquartile range,
Least upper bound and greatest lower bound of sets
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Eight questions on finding least upper bounds and greatest lower bounds of various sets.
Make decision based on expected monetary value
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Given data on probabilities of three levels of success of three options and projections of the profits that the options will accrue depending on the level of success, find the expected monetary value (EMV) for each option and choose the one with the greatest EMV.
Perform chi-squared test for differences in preferences
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Uses the $\chi^2$ test to see if there is any significant difference in preferences.
Regression Practice 1
Ready to use
Exam (2 questions) in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Statistics and probability. 2 questions. Both simple regression. First with 8 data points, second with 10. Find $a$ and $b$ such that $Y=a+bX$ . Then find the residual value for one of the data points.
Apply extended Euclidean algorithm to find gcd
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 1 other

Find gcd g of two positive integers x, y and also find integers a, b such that ax+by=g with prescribed intervals for a and b.
Find all numbers with given value of $\phi(n)$
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Given $\frac{a}{b} \in \mathbb{Q}$ for suitable choices of $a$ and $b$ , find all $n \in \mathbb{N}$ such that $\phi(n)=\frac{a}{b}n$ .
Find the smallest $n$ with given number of divisors
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Given $m \in \mathbb{N}$ , find the smallest natural number $n \in \mathbb{N}$ with $\tau(n)=m$ divisors.
Solve three congruences with the Chinese Remainder Theorem
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Solving three simultaneous congruences using the Chinese Remainder Theorem:

$\begin{eqnarray*} x\;&\equiv&\;b_1\;&\mod&\;n_1\\ x\;&\equiv&\;b_2\;&\mod&\;n_2\\x\;&\equiv&\;b_3\;&\mod&\;n_3 \end{eqnarray*}$ where $\operatorname{gcd}(n_1,n_2,n_3)=1$
Find determinant of a 3x3 matrix by row reduction
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 1 other

Given a 3x3 matrix with very big elements, perform row operations to find a matrix with single-digit elements. Then reduce that to an upper triangular matrix, and hence find the determinant.
Find determinant of a 3x3 matrix with a free variable by row reduction
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics and 1 other

Given a 3x3 matrix with very big elements, perform row operations to find a matrix with single-digit elements. Then reduce that to an upper triangular matrix, and hence find the determinant.
Find sample standard deviations of two samples
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Two ordered data sets, each with 10 numbers. Find the sample standard deviation for each and for their sum.
Multiple and partial correlation(old)
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Multiple correlation question. Given the correlation coefficent of $Y$ with $X_1$ is $r_{01}$ , the correlation coefficent of $Y$ with $X_2$ is $r_{02}$ and the correlation coefficent of $X_1$ with $X_2$ is $r_{12}$ then explain the proportion of variablity of $Y$ . Also find the partial corr coeff between $Y$ and $X_2$ after fitting $X_1$ and find how much of the remaining variability in $Y$ is explained by $X_2$ after fitting $X_1$ .
Perform t-test given sample mean and standard deviation
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

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.
Probability - draw without replacement
Doesn't work
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

A box contains $n$ balls, $m$ of these are red the rest white.

$r$ are drawn without replacement.

What is the probability that at least one of the $r$ is red?
Probability - sum of two numbers drawn without replacement
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

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.
Sample standard deviation
Draft
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Two ordered data sets, each with 10 numbers. Find the sample standard deviation for each and for their sum.
Solve a separable first order ODE with trig functions,
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Find the solution of a first order separable differential equation of the form $a\sin(x)y'=by\cos(x)$ .
Represent a linear map as a matrix given a basis
Ready to use
Question in Content created by Newcastle University by Newcastle University Mathematics and Statistics

Let $P_n$ denote the vector space over the reals of polynomials $p(x)$ of degree $n$ with coefficients in the real numbers. Let the linear map $\phi: P_4 \rightarrow P_4$ be defined by: $\phi(p(x))=p(a)+p(bx+c).$ Using the standard basis for range and domain find the matrix given by $\phi$ .

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