Transcript Sample size determination

Sample Size Determination
Population A: 10,000
Sample 10%
Sample size 1000
Population B: 5,000
Sample 15%
Sample size 750
Sampling
 The process of obtaining information from a subset (sample)
of a larger group (population)
 The results for the sample are then used to make estimates
of the larger group
 Faster and cheaper than asking the entire population
 Two keys
1. Selecting the right people
 Have to be selected scientifically so that they are
representative of the population
2. Selecting the right number of the right people
 To minimize sampling errors I.e. choosing the wrong
people by chance
Selecting the right number of the right people
Three Issues
1. Financial
2. Managerial
3. Statistical
Cost of research
Generally, the larger the sample size the smaller the
statistical error, but the greater the cost, both financial
and in terms of managerial resources
SubGroups
Male
Female
Totals
<35
100
100
200
35+
100
100
200
Totals
200
200
400
The number of subgroups to be analyzed will have an impact on
the size of the sample needed.
As the number of subgroups increases the sampling error
increases and it becomes harder to tell whether differences
between two groups are real or due to error
Determining sample size
Balance between financial and statistical issues
1. What can I afford
A critical factor will be the size
of the expected difference or
2. Rule of thumb
change to be measured, The
past experience
smaller it is, the larger the
historical precedence
sample needs to be.
gut feeling
some consideration of sample error
3. Make up of sub-groups (cells)
What statistical inferences do you hope to make between sub groups
(rare to fall below 20 for a sub group)
4. Statistical Methods
Statistical determination
Three Pieces of Information Required
1. An estimate of the population Standard
Deviation
2. The Acceptable Level of Sampling Error
3. The Desired Level of Confidence that the
Sample Result will fall within a certain range
(result +/- sampling error) of true population
values
Normal Distribution

-

a
b

The height of a normal distribution can be uniquely specified
mathematically in terms of two parameters: the mean () and the
standard deviation ().
IQ
The total area under the curve is equal to 1.
I.e. It takes in all observations
The area of a region under the normal distribution between any two
values equals the probability of observing a value in that range when
an observation is randomly selected from the distribution
For example, on a single draw there is a 34% chance of selecting
from the distribution a person with an IQ between 100 and 115
Normal Distributions
 Curve is basically bell shaped
from -  to 
 symmetric with scores
concentrated in the middle (i.e. on
the mean) than in the tails.
Mean, medium and mode
coincide
They differ in how spread out
they are.
Standard Normal Distribution (z)
Any normal distribution can be converted into a standard normal
distribution by a simple transformation formula.
Z= value of the variable – Mean of variable/SD of the variable
The mean always = zero; standard deviation always equal to one.
The probabilities in the tables are always based on a normal
distribution
Area Under Standard Normal Curve for Z values (Standard
deviations) of 1, 2 and 3
Z values
(Standard
deviations)
Area Under Standard
Normal Curve %
+/- 1
68.26
+/- 2
95.44
+/- 3
99.74
Population Vs. Sample
Population of Interest
Population
Sample
Sample
Parameter
Statistic
We measure the sample using statistics in order to draw
inferences about the population and its parameters.
Population Mean = μ Standard Deviation 
Sample Mean = X
Standard Deviation S
Sampling Distribution of the Mean
Necessary for understanding the basis for computing
sampling error for simple random samples.
A conceptual and theoretical probability distribution of the
means of all possible samples of a given size drawn from a
given population
i.e. A distribution of sample means.
If you take a sample of 100 from a population of 1000 there
are are thousands of different subsets of the population that
can be drawn, each sample will have a slightly different mean.
Those means will have also have a distribution.
Central Limit Theory says that that distribution will
approximate a normal distribution the larger the number of
samples drawn
Suppose you conducted a research study
• Took a random sample of n=100 subjects
• They tasted the new "Guacamole Doritos”
• They rated the flavor of the chip on the
following scale:
1
Too
Mild
2
3
4
5
Perfect
Flavor
6
7
Too
Hot
Results show : x1 = 2.3 and S1= 1.5
• Can you conclude that on average the target
population thought the flavor was mild?
• Suppose you take a series of random
samples of n=100 subjects:
x2 = 3.7 and S2 = 2
x3 = 4.3 and S3 = 0.5
x4 = 2.8 and S4 = .97
..
.
x50 = 3.7 and S50 = 2
The Sampling Distribution
The means of all the samples will have their own
distribution called the sampling distribution of the means
It is a normal distribution
The mean of the sampling distribution of the mean equals
the population parameter
X = (ΣXi)/n
Sampling Distribution
The standard deviation of the sampling distribution is
called the sampling error of the mean
p= π(1-π)/n
Often the population standard deviation  is
unknown and has to be estimated from the sample
S =   Σ(Xi-X)/n-1
Population distribution of the Doritos’ flavor (X)

X

Sample distribution of the x Doritos’ flavor
x
1 2
3 4
5
6
7
• What relationship does the Population Distribution
have to the Sample Distribution?
The Central Limit Theorem
Let x1, x2….. xn denote a random sample selected from a
population having mean  and variance 2. Let X denote
the sample mean. If n is large, the X has approximately a
Normal Distribution with mean  and variance 2/n.
• The Central Limit Theorem does not mean that the
sample mean = population mean.
• It means that you can attach a probability to that value
and decide.
The sampling distribution of the mean for simple random
samples that are over 30 has the following characteristics
1. The distribution is a normal distribution
2. The distribution has a mean equal to the population mean
3. The distribution has a standard deviation (the standard
error of the mean ) equal to the population standard
deviation divided by the square root of the sample size
 =  / n
X
Note: The statistic is referred to as the standard error of the
mean instead of the standard deviation to indicate that it
applies to a distribution of sample means rather than the SD of
a sample or of the population
Sampling Distribution of Proportions
We are often interested in estimating proportions or percentages
rather than means
Is the sample proportion representative of the population
proportion
The percentage of the population that has used the product
The percentage of the population that has purchased over the
Internet in the last month
The proportion of men who read a particular magazine
The sampling distribution of the proportion approximates a
normal distribution
The mean proportion of all possible samples is equal to the
population proportion
The standard error of a sampling distribution can be calculated
In practice we want to make inferences from our sample
about the population it was drawn from
What is the probability that our sample of any given
size will produce an estimate that is within one standard
error (plus or minus) of the true population
The answer is 68.26% that any one sample from a
particular population will produce an estimate of the
population mean that is within +/- one standard error of
the true value.
This is because 68.26% of all sample means from a
given population fall in this range
There is a 95.44% probability that the mean from any
one sample will within +/- two SDs
Sampling Distribution of Means
Point Estimates
 The sample mean is the
best point estimate of a
population mean
The sample mean is most likely to be close to the population
mean, but could be any of the means on the left – including one
that is a far distance from the population mean.
The distance between the sample mean and the population mean
is the sampling error
Only a small percentage of samples will have the same mean as
the population (I.e. a sampling error of zero)
Interval Estimates
Interval estimates are preferred
An interval estimate is a range of all values within which
the true population mean is estimated to fall
Normally state the size of the interval, plus the
probability that the interval will include the true
population mean.
The probability is called the confidence level (e.g. 95%)
And the Interval is called the confidence interval (e.g.
between 72 and 98)
Sample Confidence
“Probability” we can take results as “accurate
representation” of universe
(i.e. that “sample statistics” are generalisable to the real
“population parameters”)
Typically a 95% probability
(i.e. 19 times out of 20 we would expect results in this range)
Example:
We can be 95% sure
that, say, 65%
of a target market
will name Martini’s “V2” vodka
in an unprompted recall test
plus or minus 4%
We can be 95% sure (level of confidence)
that, say, 65% (predicted result)
of a target market (of a given total population)
will name Martini’s “V2” vodka
in an unprompted recall test
plus or minus 4% (to a known margin of error)
95% confidence
If we do the same test 20 times then it is
statistically probable that the results will fall
between 61-69 %, (i.e. 65 +/ 4%) at least 19
times
If we lower the probability then we lower the
sample error
e.g.. at a 90% confidence level, result might be
between 64% - 66%
(a tighter range but we are less sure the sample
is representative of the real population)
Implications for sample size
(Given reliability and validity hold)
Above a certain size little extra information is
gathered by increasing the sample size.
Generally, there is no relationship between the size of a
population and the size of sample needed to estimate a
particular population parameter, with a particular
error range and level of confidence.
To determine Sample Size we need three pieces
of information
1. The acceptable level of sampling error
2. The acceptable level of confidence
3. The estimate of the population standard
deviation
Sample Size Determination
• 3 Statistical Determinants of Sample Size
 DEGREE OF CONFIDENCE
– Statistical Confidence
– 95% Confidence or .05 Level of Significance

DEGREE OF PRECISION
– Accuracy in Estimating Population Proportion
–
+/- $5.00 versus +/- $1.00
–
+/- 10% versus +/- 5%

VARIABILITY IN THE POPULATION
– To What Degree do the Sampling Units Differ
We can choose an error range (e.g. + 5%)
We can set a confidence level (e.g. 95%)
But
Without knowing the spread of results (i.e. the standard
deviation for the population) we cannot work out the
sample size required
So
How can we estimate the population standard deviation
before selecting the sample:
• pilot tests
n = Z2σ2
• guess
E2
• previous experience Z = level of confidence
• Secondary data
σ = population SD
E = acceptable amount of sampling
error
Example
Number of fast food restaurant visits in past month
We need our estimate to be within 1/10 (.01) of a visit
from the population average (E)
We need to be 95.44% confident that the true
population mean falls in the interval defined by the
sample mean plus or minus E (i.e. within 2 standard
deviations) Z=2
Standard deviation – guess at 1.39 days
n = Z2σ2
E2
= 22(1.39) 2
(01) 2
= 4(2.93) 2 = 7.72
.01
.01
= 772
Sample Size Determination
To be
More confident
More precise
If more variable
Sample size must increase
Too big - it’s a waste of money
Too small - you cannot make a big decision
Significance level
In hypothesis testing, the significance level is the criterion used for
rejecting the null hypothesis.
The significance level is used as follows: First, the difference between
the results of the experiment and the null hypothesis is determined.
Then, assuming the null hypothesis is true, the probability of a
difference that large or larger is computed.
Finally, this probability is compared to the significance level. If the
probability is less than or equal to the significance level, then the null
hypothesis is rejected and the outcome is said to be statistically
significant.
Traditionally, experimenters have used either the .05 level
(sometimes called the 5% level) or the .01 level (1% level), although
the choice of levels is largely subjective.
The lower the significance level, the more the data must diverge
from the null hypothesis to be significant.
Therefore, the .01 level is more conservative than the .05 level.
The Greek letter alpha is sometimes used to indicate the significance
level.
Critical value
A critical value is the value that a test statistic must exceed in
order for the the null hypothesis to be rejected.
For example, the critical value of t (with 12 degrees of freedom
using the .05 significance level) is 2.18.
This means that for the probability value to be less than or equal
to .05, the absolute value of the t statistic must be 2.18 or greater.
critical value
Significance level (.05)
Test statistic
/2
/2
-2.023
0
2.023 2.816
The t distribution
The t distribution is used instead of the normal distribution
whenever the standard deviation is estimated.
The t distribution has relatively more scores in its tails than
does the normal distribution.
The shape of the t distribution depends on the degrees of
freedom (df) that went into the estimate of the standard
deviation.
As the degrees of freedom increases, the t distribution
approaches the normal distribution.
With 100 or more degrees of freedom, the t distribution is
almost indistinguishable from the normal distribution.