1. Sample Size Determination

2. Learning Objectives
To learn the financial and statistical issues in the determination of sample size
To discover methods for determining sample size
To gain an appreciation of a normal distribution
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3. Learning Objectives
To understand population, sample, and sampling distributions.
To distinguish between point and interval estimates
To recognize problems involving sampling means and proportions.
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4. Determining Sample Size for Probability Samples
The financial and statistical issues in the determination of sample size
Financial, Statistical, and Managerial Issues
As a general rule:
The larger the sample, the smaller the sampling error.
Larger samples cost more on a liner basis; however, the sampling error decreases at a rate equal to the square root of the relative increase in sample size.
Before trying to determine the size of the sample, the confidence intervals need to be decided.
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5. Determining Sample Size for Probability Samples
The financial and statistical issues in the determination of sample size
Budget Available
Sample Size—a project is often determined by the available budget
Alternative Data Collection Approaches
budget constraints force the researcher to explore and consider the value of information in relation to its cost
Rules of Thumb
Desired sampling error - Similar Studies
Past experience - A gut feeling
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6. Determining Sample Size for Probability Samples
The financial and statistical issues in the determination of sample size
Number of Subgroups To Be Analyzed
Subgroups
Samples size
Minimum needs
Traditional Statistical Methods
An estimate of the population standard deviation.
The acceptable level of sampling error.
The desired level of confidence that the sample will fall within a certain range of the true population values.
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7. The Normal Distribution
To gain an appreciation
of a normal distribution
The Normal Distribution
General Properties for the Normal Distribution
Crucial to Classical Statistical Inference
Reasons For Its Importance
Many variables have probability distributions that are close to the normal distribution
Central Limit Theorem—distribution of a large number of sample means or sample proportions will approximate a normal distribution, regardless of the distribution of the population from which they were drawn
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8. The Normal Distribution
To gain an appreciation
of a normal distribution
The Normal Distribution
Important Characteristics of the Normal Distribution
The normal distribution is bell-shaped and has only one mode.
Symmetrical about the mean
Uniquely defined by its mean and standard deviation.
The total area is equal to one.
The area between any two values of a variable equals the probability of observing a value in that range when randomly selecting an observation from the distribution.
The area between the mean and a given number of standard deviations from the mean is the same for all normal distributions
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9. The Normal Distribution
To gain an appreciation
of a normal distribution
The Normal Distribution
The Standard Normal Distribution
The same features as any normal distribution.
The mean is equal to zero
The standard deviation is equal to one.
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10. X = value of the variable  = mean of the variable
To gain an appreciation
of a normal distribution
The Normal Distribution
value of the variable - mean of the variable Z=
standard deviation of the variable
X - 
Z = 
where
X = value of the variable
 = mean of the variable
 = standard deviation of the variable
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11. To understand population, sample, and sampling distributions.
Population and Sample Distributions
To understand population, sample,
and sampling distributions.
Purpose of Conducting a Survey
To make inferences
Sample
Population Distribution
A frequency distribution of all the elements of a population. It has a mean (µ)
Sample Distribution
A frequency distribution of all the elements of an individual sample.
The mean x
The standard divergence—S
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12. Sampling Distributions To understand population, sample,
Of The Mean
To understand population, sample,
and sampling distributions.
If the samples are sufficiently large and random, the resulting distribution of sample means will approximate a normal distribution.
The distribution of the means of a large number of random samples taken from virtually any population approaches a normal distribution with a mean equal to  and a standard deviation equal to:
s x =  n √
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13. Sampling Distributions To understand population, sample,
Of The Mean
To understand population, sample,
and sampling distributions.
Basic Concepts
Characteristics of a Large Simple Random Sample
30 or more observations
Distribution is a normal distribution
Distribution has a mean equal to the population mean
Distribution has a standard deviation, referred to as the standard error of the mean, equal to the population standard deviation divided by the square root of the sample =  n √
 x
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14. Sampling Distributions To understand population, sample,
Of The Mean
To understand population, sample,
and sampling distributions.
Making Inferences on the Basis of a Single Sample
A 68.26% probability that a random sample will produce an estimate of the population mean within one standard error (±) of the true mean
95.44 % probability that a random sample will produce an estimate of the population mean within two standard error (±) of the true mean
99.74 % probability that a random sample will produce an estimate of the population mean within three standard error (±) of the true mean
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15. Sampling Distributions To distinguish between point
Of The Mean
To distinguish between point
and interval estimates.
Point Estimates
Inferences regarding the sampling error associated with a particular estimate of the population value.
Sampling Error
The distance between the x and Χ
Interval Estimate
The likelihood that a population value falls within a certain range.
Confidence Interval
The interval that in all probability, includes the true population value
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16. Sampling Distributions To distinguish between point
Of The Mean
To distinguish between point
and interval estimates.
Method of Deriving Interval Estimates
Draw a random sample from the population interest and calculating the mean of that sample
There is a 68.26% probability that the sample mean lies within one σ x
Below is the statement symbolically
1 x x
<  +
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17. Sampling Distributions Of The Mean
To recognize problems involving sampling means and proportions.
Interest in Estimating Proportions or Percentages
Rather than or in addition to estimating means. Examples
The percentage of the population that is aware of a particular
The percentage of the population that accesses the Internet one or more times in an average week
The percentage of the population that has visited a fast-food restaurant four or more times in the past 30 days
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18. Sampling Distributions
Of The Mean
To recognize problems involving sampling means and proportions.
Sampling Distribution of the proportion
A relative frequency distribution of the sample proportions of a large number of random samples of a given size drawn from a particular population.
Approximates a normal distribution
The mean proportion is equal to the population proportion.
Standard error computed as:
P (1-P) Sp = n √
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19. To recognize problems involving sampling means and proportions.
Determining
Sample Size
To recognize problems involving sampling means and proportions.
Problems Involving Means
The formula for calculating the required sample size for problems that involve the estimation of a mean: n =
Z2 2 E2
where:
Z = level of confidence expressed in standard errors
 = population standard deviation
E = acceptable amount of sampling error
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20. To recognize problems involving sampling means and proportions.
Determining
Sample Size
To recognize problems involving sampling means and proportions.
Information needed to compute the sample size
Allowable sampling error
Population standard deviation
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21. To recognize problems involving sampling means and proportions.
Determining
Sample Size
To recognize problems involving sampling means and proportions.
Four Methods Used to Deal with the Estimate of the Population Standard Deviation
Use results from a prior survey
Conduct a pilot survey
Use secondary date
Use judgment
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22. To recognize problems involving sampling means and proportions.
Determining
Sample Size
To recognize problems involving sampling means and proportions.
Problems Involving Proportions
If there is no basis for estimating P, the researcher can make the most pessimistic case regarding the value of P
Given the values of Z and E, what value of P will require the largest possible sample
A value of .50 will make the value of the expression P(1-P) larger than any possible value of P.
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23. To recognize problems involving sampling means and proportions.
Determining
Sample Size
To recognize problems involving sampling means and proportions.
Population Size and Sample Size
Make an adjustment in the sample size if the sample size is more than 5 percent of the size of the total population.
 x =  n √
N - n
N - 1
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24. To recognize problems involving sampling means and proportions.
Determining
Sample Size
To recognize problems involving sampling means and proportions.
Finite Population Correction (FPC)
An adjustment in cases where the sample is expected to be equal to 5 percent or more of the total population. (N-n) / (N-1) nN
N + n -1
n' =
where:
n' = revised sample size
n = original sample size
N = population size
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25. To recognize problems involving sampling means and proportions.
Statistical Power
To recognize problems involving sampling means and proportions.
Types of Errors
Type I Error
The error of concluding that there is a difference when there is no difference
The formulas in this chapter focus on this type of error
Type II Error
The error of saying that there is no difference when there is a difference
Statistical Power
The probability of not making a Type II error
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26. Determining Sample Size for Probability Samples
SUMMARY
Determining Sample Size for Probability Samples
Methods for Determining Sample Size
The Normal Distribution
Population, Sample, and Sampling Distributions
Sampling Distribution of the Mean
Sampling Distribution of the Proportion
Sample Size Determination
Statistical Power
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27. The End
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