You can combine a prior belief with new data to get an updated belief, and read the result as a whole distribution.
Weeks 13-14
Mathematical Statistics course units
Work in order for a complete course, or jump directly to the unit you need for an upcoming assessment.
01
Sampling distributions
Weeks 1-2
You can describe how a statistic bounces around from sample to sample, and give its centre and spread. You can also keep a population parameter and a sample statistic apart.
State the mean and standard error of the sample mean.
Recognise a chi-square or t distribution from the setting it came from.
Explain why bigger samples give a narrower sampling distribution.
Worked example
Population mean 50, standard deviation 12, samples of size 36. Describe the sample mean.
Its centre is the population mean, 50
Standard error = 12 / 6
AnswerCentred at 50 with a standard error of 2
Most common mistakeQuoting the population standard deviation as the sampling spread: using 12 instead of 2 makes every interval six times wider than it should be.
You can build a range for an unknown quantity at a chosen confidence level, with z or with t. You can also say which of the two your sample calls for.
Use t when the population standard deviation is unknown and n is small.
Find the degrees of freedom as n - 1.
Work out the sample size needed to hit a target margin of error.
Worked example
n = 25, sample mean 60, sample sd 10, and t* = 2.064 for 95%.
Standard error = 10/5 = 2
Margin = 2.064 x 2 = 4.13
Answer55.87 to 64.13
Most common mistakeUsing z* = 1.96 with a small sample and an unknown standard deviation: the interval comes out 56.08 to 63.92, too narrow to really carry 95% confidence.
You can state a null and an alternative, compute a test statistic and p-value, and reach a decision. You can also name what a Type I and a Type II error would cost.
Write both hypotheses before you look at the data.
Compare a p-value with alpha and state the conclusion in context.
Explain what a false alarm would mean in the situation at hand.
Worked example
Test H0: the mean is 100. Sample mean 106, sigma 15, n = 25.
Standard error = 15/5 = 3
z = (106 - 100)/3 = 2
AnswerTwo-sided p is about 0.046, so reject H0 at the 0.05 level
Most common mistakeReading p = 0.046 as 'a 4.6% chance the null is true': it is the chance of data this extreme assuming the null already holds.
You can write down how likely your data is for each possible parameter and pick the value that makes it most likely. You can also take logs to make the algebra easy.
Write the likelihood for a binomial sample.
Take the log, differentiate, and solve for the maximum.
Run a likelihood ratio test between two nested models.
Worked example
7 heads in 10 flips. Find the most likely value of p.
Likelihood is proportional to p to the 7 times (1 - p) cubed
Take logs and differentiate: 7/p - 3/(1 - p) = 0
Answerp = 0.7
Most common mistakePushing p to 1 because a bigger p raises p to the 7: the (1 - p) cubed factor collapses to zero, killing the whole likelihood.
You can fit a least-squares line, test whether its slope is real, and check the assumptions behind it. You can also tell a confidence interval from a prediction interval.
Test whether the slope differs from zero with a t statistic.
Read a residual plot for even spread and no pattern.
Separate an interval for the mean response from one for a single new case.
Worked example
Fitted slope 2.4, standard error 0.6, n = 20. Is the slope real?
t = 2.4 / 0.6
t = 4.0 with 18 degrees of freedom
Answerp is under 0.001, so the slope is significantly different from zero
Most common mistakeJudging the slope by its size alone: a slope of 2.4 with a standard error of 3.0 gives t = 0.8 and no evidence at all, even though 2.4 still looks big.
You can combine a prior belief with new data to get an updated belief, and read the result as a whole distribution. You can also report a credible interval.
Update a Beta prior with binomial data.
Report a 95% credible interval and say plainly what it means.
Compare the Bayesian answer with the frequentist one on the same data.
Worked example
Prior Beta(2, 2), then 7 heads and 3 tails.
Add successes to the first parameter and failures to the second
Beta(2 + 7, 2 + 3) = Beta(9, 5)
AnswerPosterior mean 9/14, about 0.643
Most common mistakeAdding the counts to the wrong parameters: Beta(5, 9) gives a mean of about 0.357, which is the chance of tails rather than heads.
Before timing yourself, check whether you can explain Sampling distributions from a blank page. Then connect it to Point estimation. If either explanation depends on copying a formula, review the unit first and complete two untimed examples.
Use tools to verify, not to choose the method for you
The Statistics calculator can test calculations and representations used in Mathematical Statistics. Make the setup yourself, predict the sign or scale, and compare the tool result with that prediction. Use the formula library to check conditions as well as notation.
Know when to move to the full test
Move from Mathematical Statistics practice to the complete course test after you can correct a missed problem without reopening the worked answer. Record the earliest wrong decision—not only the final score—so the next study session has a precise target.
Before and after the syllabus
Learn the ideas, then practise them
The unit list tells you what is covered. These two pages are where the method is explained and where you find out whether it stuck.
Questions about the Mathematical Statistics course
Where should I start?
Start with Sampling distributions if you are following the full sequence. If that unit feels automatic, open the Mathematical Statistics practice page, choose mixed review, and let the first errors identify the earliest prerequisite to revisit.
How do I know I am ready for the course test?
You are ready when you can choose a method without a hint, show the governing steps, and explain why the result is reasonable. Use the complete Mathematical Statistics test only after you can correct practice errors from a blank page.
Which calculator supports this course?
The Statistics calculator supports the calculations and representations used in this course. Use it to test or visualize a result after making your own setup.
