Statistics 2 - S2

Página 1

Chapter 1 - Binomial Distribution

When Is It Used?

There are a fixed number of trials, n.

Each trial can be only be a success or a failure.

Each trial is independent of one another.

The probability of success is constant.

Mean and Variance

To calculate the mean:

E(x) = np:

n is the number of trials.

p is the probability of success.

To calculate the variance:

Var(x) = np(1-p).

General Rules

The Binomial Distribution can only be calculated using equal-to equalities:

If the phrase is 'greater than 5' - this means X > 5:

However, in terms of probability this will go to:

1 - P(X <= 5)

Página 2

Chapter 2 - Poisson Distribution

When Is It Used?

When there is a rate of something occurring, such as the rate of car crashes or cakes being sold.

Each event must occur singly in space or time.

Each event must be independent of each other.

Each event must occur at a constant rate - the mean is proportional to the length of the interval.

Each event is said to occur randomly.

Mean and Variance

Finding the mean:

E(x) = λ

Find the variance:

Var(x) = λ

Where λ is the parameter of the distribution.

Approximating Binomial Using Poisson

When n is very large.

When p is very small.

X ¬ B(n,p), then Y - Po(np).

General Rules

REMEMBER - a rate is defined within a specific time frame, for example a week:

Watch out for questions where you will be asked the probability of something happening in 3 weeks - multiply the parameter by 3!

Página 3

Chapter 3 - Continuous Random Variables

Probability Density Function

f(x) >= 0, as we can't have negative probabilities.

P(a < X < b) = integral of f(x) with upper limit b, lower limit a.

The integral of f(x) from negative infinity to infinity is 1, as probability can't exceed 1.

Cumulative Distribution Function

F(x) = P(X <= x) = integral of f(x) with lower limit negative infinity, upper limit x.

Finding f(x) of a function:

f(x) is the differential of F(x).

Mean and Variance

Calculating the mean:

E(x) is the integral of xf(x) from negative infinity to infinity.

Calculating the variance:

Var(x) = E(X²) - E(X)²

Var(x) is the integral of x²f(x) from negative infinity to infinity.

Infinites are replaced by the ranges of the functions.

Mode, Median and Quartiles

Finding the mode:

The mode is the value of the variable where it is most dense - where the p.d.f reaches its highest point.

Finding the median:

F(m) = 0.5.

Solve to find m.

Finding the lower quartile:

F(Q1) = 0.25.

Solve to find Q1.

Finding the upper quartile:

F(Q2) = 0.75.

Solve to find Q2.

Página 4

Chapter 4 - Continuous Uniform Distribution

Probability Density Function

The probability is the same and denoted by:

1/(b-a).

Cumulative Distribution Function

x-a/b-a.

You can also use the techniques described in chapter 3.

Mean and Variance

E(X) = (a+b)/2

Var(X) = (b-a)²/12

You can also use the techniques described in chapter 3.

Página 5

Chapter 5 - Normal Approximations

Continuity Correction

As both the Poisson and Binomial distributions are both discrete, but the normal distribution is continuous, the values must be changed.

Decrease or increase the value in the probability by 0.5.

If P(X<= n), then P(Y < n + 0.5)

If P(X >= n), then P(Y >= n - 0.5)

Approximating Binomial

If n is large.

If p is close to 0.5.

X can then be approximated by:

Y - N(np, np(1-p)).

Approximating Poisson

If λ is large, then X can be approximated by:

Y - N( λ , root λ ²)

Página 6

Chapter 6 - Populations and Samples

Populations, Censuses and Samples

A population is a collection of individual items, such as plants, people or supermarkets:

A population may be of finite size - you can count the amount of items in any given population.

A population may be considered as infinite if it is impossible to know how many members there are in a population.

A population may be of countably infinite size if we know that population could be infinite, but it is countable:

The number of throws of a dice to obtain a 6.

If information is to be obtained about a population, this is known as a census.

Census:

Advantages:

Each member of a population is used.

It is unbiased.

It givens an accurate answer.

Disadvantages:

It takes a long time to do.

It is costly.

It is often difficult to ensure the whole population is surveyed.

Sampling:

For a large population where a census has been ruled out as being impractical, the information is obtained from a sub-set of a population - these members are the sample.

An investigation using sampling is called a sample survey.

The individual units of a population are known as sampling units.

When the sampling units of a population are individually named or numbered, to form a list:

This creates a sampling frame.

Sampling frames can be in the form of:

Lists.

Maps.

Files.

Databases.

Advantages:

The sample is representative of a large, whole population.

Generally much cheaper.

Saves all items in a population from being destroyed if testing results in their destruction.

Data is more readily available.

Disadvantages:

There is an uncertainty about sampling due to natural variation between samples due to the natural variation of each sampling unit.

It can be biased, as a sample could have the possibility of selecting a specific type of person, therefore not being representative:

Bias can occur if you sample from an incomplete sampling frame.

Bias can occur when you only get responess from people that have a particular interest in the topic being investigated.

Bias can occur if the person taking the sample allows their personal feelings to influence them.

Simple Random Sampling:

A sample, of size n, that is taken so that every possible sample of size n has an equal chance of being selected.

In this chapter, and the next, it is assumed that all populations are numerical and that simple random sampling is used.

A simple random sample of size n consists of:

X1, X2, X3... XN:

Xs are independent variables.

Have the same distribution as the population.

Statistics

Any characteristic of a population which is measurable is called a population parameter.

A statistic is a quantity calculated solely from the observations in a sample:

It does not involve any unknown parameters.

Examples of statistics are:

The mean or variance of a population.

Sampling Distribution

Gives all the values of a statistic and probability that each would happen by chance alone.

Created when samples from a population are repeatedly taken and the same statistic is calculated each time.

Página 7

Chapter 7 - Hypothesis Testing

Concepts

A hypothesis is a statement made about the value of a population parameter that we want to test by collecting evidence in the form of a sample.

The evidence which is summarized in the form of a statistic is called the test statistic.

We need to form two hypotheses:

The null hypothesis is the hypothesis we assume to be correct unless proved otherwise.

The alternative hypothesis, which tells us about the population parameter if our assumption is shown to be wrong.

Signifiance Level

The critical region is the range of values of the test statistic that would lead to you rejecting the null hypothesis.

The values on the boundary of the critical region are called critical values.

The level of significance is the probability we take for an event to be significant enough to reject the null hypothesis.

To calculate the ACTUAL significance level of a test, add the probability of the critical regions together.

One-Tailed Tests

A one-tailed testlookes either for an increase or decrease in the value of the parameter.

If the null hypothesis = m, then the alternative hypothesis is < m or >m, depending on if the parameter is said to have increased or decreased.

A one-tailed test will have a single part to the critical region and one critical value.

Two-Tailed Tests

When the alternative hypothesis does not equal the null hypothesis - meaning that it has changed, but it could either increase or decrease.

There will be two parts to the critical region, and two critical values.

For a given significance level, the significance at each tail is half of the overall significance level.

Binomial and Poisson, or using a Normal Approximation

You will be asked to perform hypothesis tests on both the binomial and poisson distributions.

To solve these problems, you compare the probability of a given event (such as P(X<5)) to the significance level - generally, if it is smaller, then it is significant enough to reject the null hypothesis.