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Introduction to Probability : Binomial & Normal Distribution

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### Introduction to Probability : Binomial & Normal Distribution

Dr. Marvin Reid

Objectives

- Define probability and its importance in statistical theory
- Describe the associative and multiplicative rules for joint probability under statistical independence
- Describe the properties and uses of 2 probability distribution functions the normal & binomial distributions.

Review

- Populations
- samples drawn from these populations
- Methods used to summarize the data obtained from these samples
- The relation between sample and population is uncertain. Thus to make inferences about our data we need to set up mathematical models which capture this uncertainty
- The foundation of statistical models is probability theory

Probability

- Relative frequency
- Degree of belief
- Probability reasons from the population to the sample
- Probability lies between 0 and 1

Definitions

- Trial (Experiment) – any process generates a set of results
- Outcome – the result of carrying out the trial
- Event – one or more outcomes
- Marginal probability –one event occurs

Joint conditions

- Mutually exclusive if the events cannot occur simultaneously
- Independent if the occurrence of an event does not influence the probability of another event occurring

No

Addition Rule

P(A or B)=P(A)+P(B)-P(AB)

Are events mutually exclusive

yes

Addition Rule

P(A or B)=P(A)+P(B)

Joint Probability

P(AB)=P(A) x P(B)

Are events statistically independent

yes

Marginal Probability

P(A)

Probability distributions

- Many statistical methods use probability distribution
- Probability distribution is used to calculate the theoretical probability of different values occurring
- Normal distribution – continuous data
- Binomial distribution- discrete data

Normal Distribution

- Extends from –infinity to +infinity
- Height=probability density
- Area under curve=1
- Unimodal
- mean=median=mode

Standard Normal Distribution

Any normally distributed variable can be related to the standard normal distribution whose mean is zero

and standard deviation is 1. This can be done by performing the following calculation

Z is the distance along the x axis in sd units

Some uses of Normal Distribution

95%

- Used to calculate probability of values being within specified range eg 95%CI= m ± (1.96 x se)
- Used to test inferences about the difference between a single mean and a hypothesized value and the difference between two means

The Binomial Distribution

- Describes discrete data resulting from experiments called Bernoulli process
- Each experiment (trial) has only 2 possible outcomes.
- The probability of the outcome of any trial remains fixed over time.
- The trials are statistically independent.
- Example = Toss of fair Coin

Binomial Formula

- p=probability of success
- q=(1-p)=probability of failure
- r=number of successes desired
- n=number of trials undertaken

Binomial Formula

Problem

- A couple each with sickle trait have 4 children. What is the probability that two children will have sickle cell disease.
- P(SS)=0.25, q(Non-SS)=0.75, n=4.

Characteristics of the Binomial Distribution

- When p is small the binomial distribution is skewed to the right
- As p increases the skewness is less noticeable
- When p=.5 the binomial distribution is symmetrical
- When p >0.5 the distribution is skewed to the left
- As n increases binomial distribution approximates the normal distribution (np and nq>5)

Binomial statistics

Mean of a Binomial Distribution

Standard Deviation of a Binomial Distribution

Standard error of the proportion

Some uses of the Binomial distribution

- Used to calculate probability of values being within specified range eg CI
- Used to test inferences about the difference between a single proportion and a hypothesized value and the difference between two proportions

Calculating interval estimates of the proportion from large samples

Z is the appropriate percentage point of the normal distribution

Example

- Dr. McGaw-Binns surveyed 150 medical students and found that 42% of them had a sedentary lifestyle
- A) Estimate the standard error of the proportion
- B) Construct a 95% confidence interval for the true proportion of students who had a sedentary lifestyle

Solution to example

N=150, p=0.42

Solution with stata

- The stata command
- cii number probability
- cii 150 0.42

-- Binomial Exact --

Variable | Obs Mean Std. Err. [95% Conf. Interval]

-------------+-------------------------------------------------------------

| 150 .42 .0402989 .3399811 .503244

Comparing two proportions

- p1 and p2 are the proportions
- se=standard error
- p=overall proportion based on the two sample proportions

Compare calculated z with the appropriate percentage point Zαof

the normal distribution appropriate

Objectives

- Define probability and its importance in statistical theory
- Describe the associative and multiplicative rules for joint probability under statistical independence
- Describe the properties and uses of 2 probability distribution functions the normal & binomial distributions.

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