1. Probability Normal Distribution Sampling and Sample size
Benjamin Kamala

2. Probability
Proportional of times the event occurs in a long series of random numbers

3. E.g. tossing a coin once P(H) = ½, P(T) = 1/2
Tossing a dice (1,2,3,4,5,6)
P(1)=1/6, p(2)=1/6, p(3)=1/6……
Even numbers, E: 2,4,6
Prime number, O: 2,3,5
P(E)=3/6=1/2
P(O) = 1/2

4. Note Take values from 0 to 1
P= 0=event never occur, P=1=event certainly occurs
The sum of all probabilities = 1

5. Calculation
Additional rule:(A or B)=P(A)+P(B)-P(A and B)

6. Independent events
Occurrence of one event does not affect in anyway the occurrence of the other
When a coin and Dice are tossed together the occurrence of any number on the dice does not affect the occurrence of H/T
Rule: P(A and B) = P(A)*P(B)

7. If the coin and dice are tossed together:
Find the probability of getting a H and even number (E)
P(H)=1/2
P(E)=3/6=1/2
P(H and E) = ½*1/2=1/4

8. A pack of 52 playing cards: 13 spades, 13 hearths, 13 flowers, 13 Diamonds
If a two cards are picked (one at a time) in random manner what is the probability that both are Diamonds?

9. Example
If Joshua and Hassan attend Biostatistics lectures only once in a month and there is one Biostatistics lecture per week; what is the probability of finding both of them in a given lecture
What is the probability of finding either of them in a given lecture

10. Mutually exclusive events: two events never occur together
P(A and B) = 0
E.g when the coin is tossed once P(HnT) = 0

11. Distributions Continuous Probability Distribution
Also known as Gaussian distribution
Commonest
Continuous random variable
Belly shaped
Symmetrical about ū
Determined by ū and σ

12. Normal distribution Ū = 0 σ=1
Allows different population to be compared to each other

14. Characteristics
The mean, median, and mode are all in the middle of the curve.
The total area under the curve above the x-axis is 1 square unit (50% of the area to the right, 50% to the left of the mean)

15. Probability between the limits:
Ū-1σ to Ū+1σ is 0.68
Ū-1.96σ to Ū+1,96σ is 0.95
Ū-2.58σ to Ū+2.58σ is 0.99
The values 1, 1.96, 2.58 are also known as z-score

17. To calculate a z score Z = X-Ū σ
In other literature the z-score is also known as Standard Normal Deviant (SND)
Thus in any normally distributed variable with mean Ū
and standard deviation σ, probability between x1 and x2 is the same as P between SND1 and SND2

18. A study of blood pressure of African American school boys gave a distribution of systolic
blood pressure (SBP) close to the normal with µ = 105.8mm Hg and σ = 13.4mm Hg.
What percentage of boys would be expected to have SBP greater than 120 mm Hg?

19. Calculate SND = – = 1.06
13.4
Use Table of Standard normal distribution to calculate the probabilities

20. What percentage will have SBP less than 120
What proportion will have SBP between 85 and 120
Within what limits will central 99% SBP be expected

21. Suppose the average length of stay in at MMH for type 2 DM patient is 90 days with a standard deviation of 10. If it is reasonable to assume an approximately normal distribution of lengths of stay, find the probability that a randomly selected patient from this group will have a length of stay:
Greater than 80 days
Less than 70 days
Between 60 and 120 days
Greater than 130 days
Calculate the 95% Confidence interval (CI)

22. If the Mean systolic blood Pressure (SBP)values for HKMU First year students is approximately normally distributed with the mean of 110mmHg and standard deviation of 15mmHg. Find the probability that an individual picked at random from this population will have a SBP value
A. Between 100 and 110
B. Greater than 120
C. Less than 90
D. Between 95 and 115

23. Sampling
election of a subset of individuals from within a population to estimate the characteristics of the whole population
Advantages:
Faster
Cost effective
Homogeneity of data (Accuracy and quality)
Study of the whole population: Census

24. Terms
Sampling Unit: An element or set of elements considered for selection in some stage of sampling.
Sampling Frame: A list of all the sampling units in the population.
Sampling Scheme/technique: A method of selecting sampling units from a sampling frame

25. Random vs Non random
Non-random: purposive, no valid assessment ,lead to some bias
Random: representative, a sample that has all the important characteristics

26. Classification of Sampling Methods
Non-Probability Sampling (grab or opportunity)
Convenient sampling
Quota
Probability: random sampling
Simple random sampling
Systematic Sampling
Stratified Sampling
Cluster Sampling
Multistage Sampling

27. Simple Random Sampling
Units have an equal chance of being selected.
The steps :
Obtain a numbered list of all units in the study population (i.e. availability of complete sampling frame).
Decide on the size of the sample.
Select the required number of units using either the ‘lottery’ system or tables of random numbers.

28. Systematic Sampling
elements in the sample are obtained systematically.
Steps:
Obtain the sampling frame and the size of the study population N.
Decide on the sample size, n.
Calculate the sampling interval, k = N/n.
Select the first element at random from the first k units.
Include every kth unit from the frame into the sample

29. Stratified Sampling population is divided into subgroups (or strata)
each stratum is sampled randomly with a known sample size.
Strata may be defined according to some characteristics of importance in the survey (e.g. occupation, religion, age groups or even locality)

30. Stratified sampling
Units within a stratum should be as much alike as possible and units in different stratum should be as much different as possible

31. Cluster Sampling
Obtaining a complete list of individuals in the study population is not feasible or practical, or a complete sampling frame is not available before the investigation starts.
Sampling units are a collection (cluster) of study units
within a cluster should as heterogeneous as possible while the between-cluster variability should be as low as possible

32. Multistage Sampling
Multi-stage sampling is carried out in more than 1 stage, and different sampling techniques can be employed at every stage

33. Sample size Depends on three parameters Sample size:
Directly proportion to level of CONFIDENCE
Directly proportion to VARIABILITY
Inversely proportion to acceptable ERROR

34. Sample size for the mean
n = z2σ 2
ε 2
n=sample size
σ standard deviation
ε = maximum acceptable error

35. Sample size for the proportion
n = z2 Π ( 1- Π )
ε 2 Or
n = z2 Π ( 100- Π )
Where n= sample size
Π= proportion
ε = error

37. A study is planned to estimate the mean BW of babies born at MMH
A study is planned to estimate the mean BW of babies born at MMH. BW is approximately normally distributed and 95% of babies born are probably between 2000 and 4000g. Determine the sample size required so that there is 95% chance that the estimated BW does not differ from the true value by more than 50g