Download Hypothesis Testing--Calculating Sample Size

UNC-Wilmington
Department of Economics and Finance
ECN 377
Dr. Chris Dumas
Calculating Sample Size (n)
Methodology
Step 1. Choose E.
When calculating sample size, the investigator must think ahead and choose an Allowable Level of Error
(E) he or she is willing to tolerate. This decision is subjective, a judgement call. The lower the level of
error desired, the more data that will need to be collected.
E = allowable error
E = acceptable difference between sample mean and true population mean
E = sample mean – population mean
E = Xbar - μ
Think of E as the allowable “plus or minus” amount in the eventual estimate, the amount that the estimate
(Xbar) will be “plus or minus” relative to the true population value (μ).
Step 2. Choose s.
The investigator must also choose “s”, an estimate of the sample standard deviation. Now, at this point,
the investigator has no data, so, “s” is an educated guess based on prior studies, a small pilot sample, or
just an educated guess. To conduct a more conservative test, use a larger value of s.
Step 3. Choose your Confidence Level, calculate your Significance Level (α/2) and t-value
Choose your Confidence Level for your eventual test, typically 95%. Based on this, calculate your
Significance Level for a two-sided test (α/2), typically, α = 5%, and so α/2 = 2.5%. We assume a twosided situation because our eventual estimate will be “plus or minus” E from the true value, and “plus or
minus” implies that the error is two-sided. Based on this, next we find the corresponding t-value from the
t-Table. Use infinity (∞) for the degrees of freedom (d.f.) in the t-Table (this is based on the assumption
that your sample size, n, will end up being larger than 30, which is usually the case).
Step 4. Plug everything into the ttest formula, and solve for n.
𝑡𝑡𝑒𝑠𝑡 =
𝑡𝑡𝑒𝑠𝑡 =
𝑋𝑏𝑎𝑟 − 𝜇0
𝑠. 𝑒.
𝐸
𝑠
( )
√𝑛
𝑠 ∙ 𝑡𝑡𝑒𝑠𝑡 2
𝑛=(
)
𝐸
Remember: the ttest value is for α/2
Step 5. Interpretation
Sample size “n” will produce an estimate (Xbar) that is “plus or minus” E from the true population mean
(μ) with the Confidence Level you have chosen (in this example, 95%).
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UNC-Wilmington
Department of Economics and Finance
ECN 377
Dr. Chris Dumas
Example
Suppose the Business School believes that current business students are much smarter than
students from ten years ago. To test this hypothesis, they propose to give an assessment test to a
sample of the current business students and compare the mean score with the mean score from
ten years ago. If the Business School wants to estimate the mean score on the assessment test for
the current population of students +/- 5 points with 95% confidence, given that the standard
deviation of scores is approximately 10 (based on the historical average from past classes of
students), what sample size, "n," of current students is required?
𝑡∙𝑠 2
The formula for sample size is: 𝑛 = ( )
𝐸
We are given: E = 5, s = 10.
For a 95% Confidence Level ==> α = 1 - 0.95 = 0.05
Since our eventual estimate could be in error on either side, we need the TWO-SIDED
t-value, so we find the t-value for α/2 = 0.025, and we assume that n will be large, so we assume
that d.f. = ∞. Looking in the t-table for d.f. = ∞ and probability = 0.025 gives:
t = 1.96
Plugging everything into the formula gives:
𝑡∙𝑠 2
𝑛 = ( 𝐸 ) = [(1.96∙10)/5]2 = 15.37
Interpretation:
If a random sample of n = 15 students takes the assessment test, the mean test grade for the
sample (Xbar) will be “plus or minus” E = 5 points from the true population mean (μ) with a
Confidence Level of 95%.
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