revisiting sampling concepts. population a population is all the possible members of a category...
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Revisiting Sampling Concepts
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Population
• A population is all the possible members of a category
• Examples: • the heights of every male or every female• the temperature on every day since the beginning of time• Every person who ever has, and ever will, take a particular
drug
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Sample
• A sample is some subset of a population
– Examples:• The heights of 10 students picked at random• The participants in a drug trial
• Researchers seek to select samples that accurately reflect the broader population from which they are drawn.
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PopulationSample
Sample Statistics
PopulationParameters
Inference
Samples are drawn to infer something about population
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Reasons to Sample
Ideally a decision maker would like to consider every item in the population but;
• To Contact the whole population would be time consuming e.g. Election polls
• The cost of such study might be too high
• In many cases whole population would be consumed if every part of it was considered
• The Sample results are adequate
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Probability Vs Non Probability Sampling
Probability Sampling
• Drawing Samples in Random manner
• Using random numbers
• Writing names on identical cards or slips and then drawing randomly
• Choosing every nth item of the population
• First dividing the population into homogeneous groups and then drawing samples randomly
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Probability Vs Non Probability Sampling
Non Probability Sampling
• man-on-the-street interviews
• call-in surveys
• readership surveys
• web surveys
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Types of Variables
• Qualitative• Quantitative• Discrete• Continuous
• Categorical • Numerical
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Sampling Error
• “Sampling error is simply the difference between the estimates obtained from the sample and the true population value.”
Sampling Error = X - µWhere X = Mean of the Sample µ = Mean of the Population
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Validity of Sampling Process
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Sampling Distributions
• A distribution of all possible statistics calculated from all possible samples of size n drawn from a population is called a Sampling Distribution.
• Three things we want to know about any distribution?
– Central Tendency
– Dispersion
– Shape
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Sampling Distribution of Means
• Suppose a population consists of three numbers 1,2 and 3
• All the possible samples of size 2 are drawn from the population
• Mean of the Pop (µ) = (1 + 2 + 3)/3 = 2
• Variance
• Standard Deviation = 0.82
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Distribution of the Population
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Sampling distribution of means
n = 2
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Sample # Sample Sample Mean
1 1,1 1
2 1,2 1.5
3 1,3 2
4 2,1 1.5
5 2,2 2
6 2,3 2.5
7 3,1 2
8 3,2 2.5
9 3,3 3
Mean of SD 2
= µ
= 0.6
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= µ
• The population’s distribution has far more variability than that of sample means
• As the sample size increases the dispersion becomes less and in the SD
<
0.6 < 0.8
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• The mean of the sampling distribution of ALL the sample means is equal to the true population mean.
• The standard deviation of a sampling distribution called Standard Error is calculated as
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Central Limit Theorem ……
• The variability of a sample mean decreases as the sample size increases
• If the population distribution is normal, so is the sampling distribution
• For ANY population (regardless of its shape) the distribution of sample means will approach a normal distribution as n increases
• It can be demonstrated with the help of simulation.
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Central Limit Theorem ……
• How large is a “large sample”?
• It depends upon the form of the distribution from which the samples were taken
• If the population distribution deviates greatly from normality larger samples will be needed to approximate normality.
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Implications of CLT
• A light bulb manufacturer claims that the life span of its light bulbs has a mean of 54 months and a standard deviation of 6 months. A consumer advocacy group tests 50 of them. Assuming the manufacturer’s claims are true, what is the probability that it finds a mean lifetime of less than 52 months?
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Implications of CLT Cont
• From the data we know that
• µ = 54 Months = 6 Months
• By Central Limit Theorem
= µ = 54
=
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54
o-2.35
0.0094
52
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• To find ,we need to convert to z-scores:
• From the Area table = 0.4906
• Hence, the probability of this happening is 0.0094.
• We are 99.06% certain that this will not happen
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What can go wrong
• Statistics can be manipulated by taking biased samples intentionally
Examples
• Asking leading questions in Interviews and questionnaires
• A survey which showed that 2 out 3 dentists recommend a particular brand of tooth paste
• Some time there is non response from particular portion of population effecting the sampling design
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How to do it rightly
• Need to make sure that sample truly represents the population
• Use Random ways where possible
• Avoid personal bias
• Avoid measurement bias
• Do not make any decisions about the population based on the samples until you have applied statistical inferential techniques to the sample.