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# Introduction to Probability : Binomial & Normal Distribution - PowerPoint PPT Presentation

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

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### Introduction to Probability : Binomial & Normal Distribution

Dr. Marvin Reid

• 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.

• 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

• Relative frequency

• Degree of belief

• Probability reasons from the population to the sample

• Probability lies between 0 and 1

• 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

• 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

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

Are events mutually exclusive

yes

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)

• 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

• Extends from –infinity to +infinity

• Height=probability density

• Area under curve=1

• Unimodal

• mean=median=mode

y1;µ=50;σ=5

y2;µ=50;σ=10

Normal distribution-variation with sd

Completely described by the mean & sd

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

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

• 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

• p=probability of success

• q=(1-p)=probability of failure

• r=number of successes desired

• n=number of trials undertaken

Binomial Formula

• 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.

• 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)

P=0.1

P=0.5

P=0.7

P=0.4

Mean of a Binomial Distribution

Standard Deviation of a Binomial Distribution

Standard error of the proportion

• 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

Z is the appropriate percentage point of the normal distribution

Example samples

• 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 samples

N=150, p=0.42

Solution with stata samples

• 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 samples

• 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 samples

• 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.