biostatist

SAMPLING DISTRIBUTION

SAMPLING DISTRIBUTION
It is the distribution of all possible values which can be assumed by some statistic, computed from samples of the same size randomly drawn from the same population
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1 2 3 4 5
10 Samples

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 Make a table of the observed values of the statistic and its corresponding frequency For any sampling distribution we are interested in the mean , variance, and the shape of the curve.
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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 means of the _ samples (X) will be normal

DISTRIBUTION OF THE SAMPLE MEAN

DISTRIBUTION OF THE SAMPLE MEAN
The standard deviation of these means will be σ/√n _ X- µ Z=--------- σ/√n

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.The mean of the means of the samples µX will be equal to the mean of the underlying population from which these samples were drawn.The standard deviation of these means will be σ/√n

DISTRIBUTION OF THE SAMPLE MEAN

X

SE= σ/√n

DISTRIBUTION OF THE SAMPLE MEAN
X- µ Z=--------- σ/√n

Exercise

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 P=0.1379 σ/√n 12.7 / √ 10 Z(1.09)=0.8621 PZ(1.09)=1- 0.8621 = 0.1379


_ X= 190 , µ = 185.6 , σ= 12.7 , n=10 _ X- µ 190-185.6 Z=--------- =---------------- =1.09 σ/√n 12.7 / √ 10P(Z1.09)=0.8621P=1- 0.8621= 0.1379

Distribution of the difference of two sample means

_ _ (x1 _ x2)-(µ1- µ2)Z=---------------------- √ (σ21/n1)+ (σ22/n2)

Distribution of the difference of two sample means

(X1-X2)

SE= √(σ12/n1)+ (σ22/n2)

EXERCISE
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.


_ _x1 _ x2 = 50 , µ1- µ2 = 0.0 , σ21=19600, σ22=8100, n1= 15, n2= 10 _ _ (x1 _ x2) - (µ1- µ2) 50-0.0Z=------------------------- = ----------------------------- √ (σ21/n1)+ (σ22/n2) √ (19600/15)+ (8100/10) 50=------ = 1.09 P(Z1.09)=0.8621 46 P=1-0.8621= 0.1379


DISTRIBUTION OF SAMPLE PROPRTION
~ P _ PZ=------------- √ p(1-p)/n

DISTRIBUTION OF SAMPLE PROPRTION

~ P - PZ=------------- √ p(1-p)/n P
~ P
√ p(1-p)/n

EXERCISE

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%


~ P=0.15 , P= 0.08 , n=150 ~ P _ P 0.15-0.8Z=------------- = ------------------------------- √ p(1-p)/n √ 0.08(1-0.08)/ 150 0.07=------------ = 3.18 P (Z 3.18) = 0.9993 0.022P=1- 0.9993= 0.0007

DISTRIBUTION OF DIFFERENCE BETWEEN TWO SAMPLE PROPRTIONS

~ ~ (P1_P2) _ ( P1_P2)Z=---------------------------- √ p1(1-p1)/n1 + p2(1-p2)/n2

DISTRIBUTION OF DIFFERENCE BETWEEN TWO SAMPLE PROPRTIONS

~ ~ (P1-P2) _ ( P1-P2)Z=---------------------------- √ p1(1-p1)/n1 + p2(1-p2)/n2 √ p1(1-p1)/n1 + p2(1-p2)/n2 (P1-P2)
~ ~ (P1-P2)

EXERCISE

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%.

~ ~P1 = 0.1 , P2 = 0.1 , P1 - P2 = 0.06 , n1= 250 n2= 250 ~ ~ (P1_P2) _ ( P1_P2)Z=--------------------------------- √ p1(1-p1)/n1 + p2(1-p2)/n2

~ ~ (P1 _ P2) _ ( P1 _ P2)Z=--------------------------------- √ p1(1-p1)/n1 + p2(1-p2)/n2 (0.06) _ (0.1-0.1) 0.06-0.0=----------------------------------------------------------------- = ----------------------------------------------- √ 0.1(1-0.1)/250 + 0.1(1-0.1)/250 √ (0.1) (0.9) /250 + (0.1) (0.9) /250 0.06=--------- = 2.22 P(Z2.22)= 0.9864 0.027P=1-0.9864= 0.0136