PROBABILITY DISTRIBUTION

PROBABILITY DISTRIBUTIONS We usually compare our observations in the studies with a theoretical probability distribution which is described by a mathematical models. Depending on whether the random variable is discrete or continuous, the probability distribution can be either discrete or continuous.

Discrete probability distribution 1. Binomial distribution  

Discrete probability distribution 2. Poisson distribution Used to determine the probability of the rare events, when the average number of successes is known.

Continuous probability distribution we can only derive the probability of the random variable if it is continuous variable. Continuous pr. Distributions include (Normal test (z test), t test, F test and Chi squared test)

The Normal Distribution “Gaussian Distribution” It is the most important distribution in statistics The parameters in this distribution are the: Population mean (µ) as a measure of central tendency Population standard deviation (σ) as a measure of dispersion

Mode Median

The Normal Distribution “Gaussian Distribution” It is used for continuous variables The curve is symmetric around the mean The total area under the curve equal one The mean, median, and the mode are equal

The Normal Distribution “Gaussian Distribution” 50% of the area under the curve is on the right side of the curve and the other 50% is on its left. Not kurtotic and not skewed.

Skewness Refers to the degree of asymmetry of the distribution of the variable. It is skewed to the right (+ve) , it has long tail to the right with few high values ( the mean is larger than the median). It is skewed to the left(-ve) , it has long tail to the left with few low values ( the mean is smaller than the median). So the median is more stable than the mean

Kurtosis Refers to the ‘peakedness’ of the distribution. The flat curve is called platykurtotic. The peaked curve is called leptokurtotic.

Skewness

The Normal Distribution “Gaussian Distribution” µ ± 1 σ 68% of the area µ ± 2 σ 95% of the area µ ± 3 σ 99.7% of the area

The Normal Distribution “Gaussian Distribution” Different values of µ will shift the graph of the distribution along the X axis

The Normal Distribution “Gaussian Distribution” With fixed (σ) any change in (µ) will not change the shape of the curve, but it will be shifted to: the right ( when µ is increased) or to the left (when µ is decreased)

The Normal Distribution “Gaussian Distribution” Different values of (σ) determine the degree of flatness or peakedness of the graph of the distribution When (σ) is increased the curve will be more flat When (σ) is decreased the curve will be more peaked

The unit normal , or the Standard normal distribution X- µ Z= ---------- σ

Exercise 1 Find for a standard normal distribution P(0< Z <1.2) P(Z <-1.2 or Z >1.2) P(Z <1.2) P(1.5 < Z <2.0)

Looking up probabilities in the standard normal table What is the area to the left of Z=1.51 in a standard normal curve? Z=1.51 Area is 93.45% Z=1.51

Exercise 2 If µ of DBP of a population = 80 mmHg, and σ2 =100(mmHg)2 .What is the probability of selecting a man with DBP of: P(75< X < 85) P(60< X <100) P(65< X <95) P(X <60) P(X >100) P(90< X <100)

Exercise 3 If the weight of 6-years old boys are normally distributed with µ =25 Kg, and σ = 2 kg. Find: P(20< X <25) P(X >28) P(X >22) P(X <22) P(X <28) P(26< X <29)

SAMPLING DISTRIBUTION It is the distribution of all possible values of a statistic (like mean), computed from samples of the same size randomly drawn from the same population

STEPS IN CONSTRUCTING SAMPLING DISTRIBUTION From a population of size (N), we randomly draw all possible samples of size (n) From each sample we compute the statistic of interest ( usually the mean)

STEPS IN CONSTRUCTING SAMPLING DISTRIBUTION Make a table of the observed values of the statistic and its corresponding frequency For any sampling distribution(of the samples) we are interested in the mean , variance, and the shape of the curve.

DISTRIBUTION OF THE SAMPLE MEAN When sampling is from a normally distributed population , the distribution of the sampling mean will posses the following properties: The distribution of the mean of the _ samples (X) will be normal (Central Limit Theorem).

DISTRIBUTION OF THE SAMPLE MEAN The mean of the means of the _ samples X will approximately be equal to the mean of the underlying population (µ) from which these samples were drawn

DISTRIBUTION OF THE SAMPLE MEAN The standard deviation of these means will be σ/√n , n is the number of values in the sample. _ X- µ Z=--------- σ/√n

Exercise 4 If the cranial length of certain large human population is normally distributed with a mean =185.6 mm, and standard deviation=12.7 mm. What is the probability that a random sample of size 10 from this population will have a mean greater than 190 mm?

Exercise _ X- µ 190-185.6 Z=--------- =-------------=1.09 σ/√n 12.7 / √ 10 P for (Z=1.09) = 0. 8621 P (greater than 190 mm ) =1- 0.8621 =0.1379

Distribution of the difference of two sample means  

EXERCISE 5 If the level of vitamin A in the liver of two human populations is normally distributed, the variance of population 1 =19600 unit2, and of population 2 =8100 unit2. If there is no difference in population means , what is the probability of having a difference in means between two samples (n1=15, n2=10) drawn at random is equal or greater than 50 unit.

DISTRIBUTION OF SAMPLE PROPRTION  

EXERCISE 6 Suppose in a certain human population , the prevalence of color blindness is 8%. If we randomly select 150 individuals from this population, what is the probability that the prevalence in the sample is as great as 15%

DISTRIBUTION OF DIFFERENCE BETWEEN TWO SAMPLE PROPRTIONS  

EXERCISE 7 In a certain population of teenagers, it is known that 10% of boys are obese. If the same proportion of girls in the population are obese, what is the probability that a random sample of 250 boys and 200 girls will yield a difference in prevalence of 6%.

EXERCISE 8 It has been found that after a period of training , the mean time required for certain handicapped persons to perform a particular task is 25 seconds, with a standard deviation of 5 seconds. Assuming a normal distribution of times, find the probability that a sample of 25 individuals will yield a mean of:

EXERCISE 8 26 seconds or more Between 24 and 27 seconds 26 seconds or less Greater than 22 seconds

EXERCISE 9 In population A , the average annual family expenditure on general health is 380 $, with a variance of 2899 $2. In population B the comparative figures were 380 $, and 3250 $2. A random sample of 40 families from population A , and 35 families from population B, showed average annual expenditure on health of 346 $ and 300 $ ,respectively. Find the probability of such a difference or higher?

EXERCISE10 A psychiatric social worker believes that in both community A and community B , the proportion of adolescents suffering from some mental or emotional problem is 20%. In a sample of 150 adolescents from community A , 15 had mental or emotional problem. In a sample of 100 from community B, the number was 16. What is the probability of observing such a difference or higher ?