• Point Estimate and Interval Estimate
An interval estimate is an interval of numbers within which
the parameter value is believed to fall.
A point estimate is a single number that is our “best guess”
for the parameter.
• Confidence Interval
A confidence interval is an interval containing the most
believable values for a parameter. The probability that this
method produces an interval that contains the parameter is
called the confidence level. This is a number chosen to be
close to 1, most commonly 0.95
• Margin of Error
The margin of error measures how accurate the point
estimate is likely to be in estimating a parameter. It is a
multiple of the standard error of the sampling distribution of
the estimate, such as 1.96×(standard error) when the
sampling distribution is a normal distribution.
A 95% confidence interval for a population proportion p is
p
p̂ ± 1.96(se), with se = p̂(1 − p̂)/n
where p̂ denotes the sample proportion based on n observations.
Warning: we assume here the sample size is large, i.e. the number
of successes is larger than 15 and the number of failures is larger
than 15.
A 95% confidence interval for the population mean µ is
√
x̄ ± t.025,df (se), with se = s/ n
where t.025,df denotes the t-score of the t-distribution and
df = n − 1 denotes the degrees of freedom of the corresponding
t-distribution.
z-Scores for the Most Common Confidence Levels
Confidence Level
0.90
0.95
0.99
Error Probability
0.10
0.05
0.01
z-Score
1.645
1.96
2.58
Confidence Interval
p̂ ± 1.645(se)
p̂ ± 1.96(se)
p̂ ± 2.58(se)
Effects of Confidence Level and Sample Size on Margin of Error
The margin of error for a confidence interval:
• Increases as the confidence level increases
• Decreases as the sample size increases.
Sample Size for Estimating a Population Proportion
The random sample size n for which a confidence interval for
a population proportion p has margin of error m (such as
m = 0.04) is
p̂(1 − p̂)z 2
n=
m2
The z-score is based on the confidence level, such as z = 1.96
for 95% confidence. We either guess the value for the sample
proportion p̂ based on other information or take the safe
approach of setting p̂ = 0.5.
Sample Size for Estimating a Population Mean
The random sample size n for which a confidence interval for
a population mean µ has margin of error m (such as
m = 0.04) is
σ2z 2
n=
m2
The z-score is based on the confidence level, such as z = 1.96
for 95% confidence. To use this formula, we guess the value
for the population standard deviation σ.
Confidence Interval for the Population Proportion p When the
Sample Size Is Small
Suppose a random sample does NOT have at least 15 successes
and 15 failures. The confidence interval formula
p
p
p̂ − z p̂(1 − p̂)/n, p̂ + z p̂(1 − p̂)/n
still is valid if we use it after adding 2 to the original number of
successes and 2 to the original number of failures.