#### The Two Sample t - Test

If we have two independent populations with unknown variances, then we can use the t â€“distribution to test hypotheses about the means of the two populations â€“ either that they are equal or that they differ by a certain amount. The procedure is: Write...

https://astarmathsandphysics.com/a-level-maths-notes/s4/3819-the-two-sample-t-test.html#### Introduction To The Normal Distribution

The normal distribution is the most useful, adaptable, flexible, used statistical distribution there is. All other distributions are, in a sense, subsumed by it because of the Central Limit Theorem, which says that in some circumstances any statistical...

https://astarmathsandphysics.com/a-level-maths-notes/s1/3713-introduction-to-the-normal-distribution.html#### Regression Lines

The correlation coefficient can be found if necessary to found how good a relationship we have. You may be given summary statistics etc but I will illustrate an example from scratch. Example: Find the equation of the regression line of height above...

https://astarmathsandphysics.com/a-level-maths-notes/s1/3730-regression-lines.html#### Simultaneous Equations Problems Involving the Normal Distribution

set of data and so cannot find the mean and standard deviation from a set of readings. However, given JUST TWO suitable statistics we can find the mean and standard deviation of a list. This is because there are two constants we must find, and any set...

https://astarmathsandphysics.com/a-level-maths-notes/s1/3733-simultaneous-equations-problems-involving-the-normal-distribution.html#### The Goodness of Fit or Chi - Squared Distribution

The testcan only be used the goodness of fit of a data set to a hypothesisedprobability distribution. The observed data is sorted into frequency classes, and for eachfrequency class, the expected number of observations that would fallinto that...

https://astarmathsandphysics.com/a-level-maths-notes/s3/3795-the-goodness-of-fit-or-chi-squared-distribution.html#### A Level Maths Notes - S3

https://astarmathsandphysics.com/a-level-maths-notes/s3.html#### Summary of Formulae - Conditional Probability, Independent Events etc

This is only true if A and B are independent. You can never assume A and B are independent. You must either know it or prove it. This is always true for two events A and B. The left hand side P(A divides B) is the probability that A will happen given...

https://astarmathsandphysics.com/a-level-maths-notes/s1/3738-summary-of-formulae-conditional-probability-independent-events-etc.html#### The Exponential Distribution

The probability distribution for the exponential function, withprobability distribution function isillustrated below. A number of situations can be modelled by the exponentialdistribution. In these examples it must be assumed however that eachevent is...

https://astarmathsandphysics.com/a-level-maths-notes/s4/3815-the-exponential-distribution.html#### Tree Diagrams

We can represent any combination or sequence of independent events on a tree diagram. Each possibility at each stage lies on an exclusive branch of the diagram. At each stage the probability of that sequence of events equals the multiple of the...

https://astarmathsandphysics.com/79-main/4905-tree-diagrams.html#### Rutherford's Model of the Atom

An experiment which involved firing alpha particles at a piece of thin gold filed lead Ernest Rutherford to draw up a new model of the atom as a very small heavy nucleus surrounded by orbiting electrons. To the great surprise of Rutherford, and the two...

https://astarmathsandphysics.com/ib-physics-notes/the-history-and-development-of-physics/1322-rutherford-s-model-of-the-atom.html#### Error Bars

Error bars are used on graphs to indicate the error, or uncertainty in a reported measurement. They give a general idea of how accurate a measurement is, or conversely, how far from the reported value the true (error free) value might be. Error bars...

https://astarmathsandphysics.com/a-level-physics-notes/principles-dimensions-units-and-error-analysis/2979-error-bars.html#### One or Two Tailed Hypothesis Tests - Discussion

which would result in the null hypothesis beingreject. Hypothesis tests based on on probabilities or teststatistics must be conducted once for each test, but hypothesistests based on critical values can be carried out many times withlittle extra effort....

https://astarmathsandphysics.com/a-level-maths-notes/s2/3761-one-or-two-tailed-hypothesis-tests-discussion.html#### Significance Levels and p - Values

When conducting a hypothesis test, the criterion for rejecting thenull hypothesis is that an observed value is so unlikely assumingthat the null hypothesis is true that it must in fact be true. Todecide this, each hypothesis test is associated with a...

https://astarmathsandphysics.com/a-level-maths-notes/s2/3765-significance-levels-and-p-values.html#### Graphical Representation of Uncertainty

In many situations the best method to present an assess data is to use a graph. Error bars enable goodness of fit to a specific statistical model to be assessed. Since the error represents the uncertainty range, the best fit curve should pass through...

https://astarmathsandphysics.com/ib-physics-notes/measurements-units-uncertainty-and-principles/1342-graphical-representation-of-uncertainty.html#### Error Anaysis 2

Error has to do with uncertainty in measurements that nothing canbe done about. A metre rule might measure to the nearest mm. If ameasurement is repeated, the length obtained may differ and none ofthe measurements can be preferred. Although it is not...

https://astarmathsandphysics.com/university-physics-notes/experimental-physics/1560-error-analysis-2.html#### Markov Chains

The probability model for a sequence of trials in which the probability of ending any trial in a state depends on the outcome of the preceding trial is called a Markov chain. Formally: A Markov chain is a random process where the random variables are...

https://astarmathsandphysics.com/university-maths-notes/stochastic-processes/2077-markov-chains.html#### Queueing Notation

The statistical model that has notation in three parts. The first part denotes the process of arrivals at the start of a queue. Usually the process of arrival of customers is radom or 'Markovian' and denoted 'M'. The second part denotes the process by...

https://astarmathsandphysics.com/university-maths-notes/stochastic-processes/2080-queueing-notation.html#### Error Analysis 2

Error has to do with uncertainty in measurements that nothing can be done about. A metre rule might measure to the nearest mm. If a measurement is repeated, the length obtained may differ and none of the measurements can be preferred. Although it is...

https://astarmathsandphysics.com/a-level-physics-notes/principles-dimensions-units-and-error-analysis/2978-error-analysis-2.html#### Maxwell's Demon

James Clerk Maxwell introduced a now-famous thought experiment that highlighted the contrast between the statistical nature of entropy and the deterministic nature of the underlying physical processes. This experiment, known as Maxwell's demon,...

https://astarmathsandphysics.com/a-level-physics-notes/thermal-physics-and-gases/3048-maxwell-s-demon.html#### The Recurrence Paradox

The second law of thermodynamics states that energy flows from hotter to colder bodies in a way that increases the total disorder of the Universe, or equivalently, that the particles of the Universe tend to move to states that are 'more probable'. In...

https://astarmathsandphysics.com/a-level-physics-notes/thermal-physics-and-gases/3069-the-recurrence-paradox.html