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Quantitative Methods
CONFIDENCE INTERVALS
Notation
n = sample size,
C = confidence level
X̄ = sample mean
s = sample standard deviation
p̂ = sample proportion
µ = population mean
σ = population standard deviation
p = population proportion
Confidence interval for a mean
s
X̄ ± t∗ √
n
where t∗ is the critical value for the t-distribution with df = n − 1.
How to find t∗ :
Using the t-distribution table, identify the row corresponding to df = n − 1, and the column
corresponding to the confidence level (shown at the bottom). At the intersection you will find the
value of t∗ .
Confidence intervals for a proportion
r
p̂ ± z ∗
p̂(1 − p̂)
n
where z ∗ is the critical value for the normal distribution.
How to find z ∗ :
The area under the normal curve below z ∗ needs to be the confidence level C + the area of the
1−C
tail, which is 1−C
2 . So, calculate the desired area C + 2 , and look inside the normal distribution
table to find that number (or the closest number to it). The corresponding row and column will
give you the value of z ∗ .
REMARKS
• If we take a large number of samples and find a 95% confidence interval for each, then the
population mean (or proportion) would be contained in the interval 95% of the times.
• For higher confidence levels, the interval will be wider.
• The bigger the sample, the more certain we are about X (or p), and therefore the confidence
interval gets smaller.