Correct and Incorrect Ways to Explain a Confidence Interval
USATODAY / CNN / Gallup Poll 11-17-2003 Presidential Poll
Correct Confidence Interval Statements
We are 95% confident that the percentage of adult Americans who approve of the way
President Bush is handling his job is between 46.9% and 53.1%.
95% of the intervals constructed using this method will include the true percentage of
adult Americans who approve of the way President Bush is handling his job.
We are 95% confident that the percentage of adult Americans who approve of the way
President Bush is handling his job is within 3.1% of 50%.
The probability of drawing a random sample of 1,000 adult Americans and using their
survey responses to construct an interval using this method that captures the true
population percentage of Americans who approve of the way President Bush is handling
his job is .95.
Incorrect Confidence Interval Statements
The probability is .95 that percentage of adult Americans who approve of the way
President Bush is handling his job is between 46.9% and 53.1%.
There is a 95% chance that the true percentage of adult Americans who approve of the
way President Bush is handling his job falls within this interval.
There is a 95% probability that when 1,000 adult Americans are surveyed, there will be
between 46.9% and 53.1% of the respondents who approve of the way President Bush is
handling his job.
In repeated sampling, 95% of the sample percentages of adult Americans who approve of
the way President Bush is handling his job will be between 46.9% and 53.1%.
Based on Actual Student Responses to Question #6 on the 2000 AP Statistics Exam
Correct Confidence Interval Statements
We are 95% confident (or sure or certain) that the proportion of couples for which the
wife is taller than the husband is between .029 and .071.
95% of the intervals constructed using this method will include the proportion of the
married couples in the population for which the wife is taller than the husband.
We are 95% confident (or sure or certain) that the proportion of couples for which the
wife is taller than the husband is within .0214 of .05.
The probability of drawing a random sample of 400 couples for which the true proportion
of couples having wives taller than their husbands is in the interval constructed in this
method is .95.
Incorrect Confidence Interval Statements
The probability that the proportion of couples for which the wife is taller than the
husband is between 2.9% and 7.1% is .95.
There is a 95% chance that the proportion of married couples in the population for which
the wife is taller than her husband lies in this interval.
With 95% confidence, the probability that, in the population of couples. the wife is going
to be taller than the husband will range from approximately 3% to 7%.
95% of the time, the wife will be taller than the husband in 2.86% to 7.14% of the
couples.
95% of the time, the proportion of married couples in which the wife is taller than her
husband lies between .029 and .071.
There is a 95% probability that when 400 couples are tested, there will be between 2.9%
and 7.1% of the couples in which the wife is taller than the husband.
In repeated sampling for the proportion of couples in which the wife is taller than the
husband, 95% of the proportions will be between .029 and .071.
Correct Interpretation of a Confidence Interval
We are 95% [or actual value from the context of the problem if different from 95]
confident that the true [population parameter from context of the problem] is between
[lower bound estimate] and [upper bound estimate].
Correct Interpretations of a Confidence Level
A 95% [or actual value from the context of the problem if different from 95]
confidence level means that if we took repeated simple random samples of the same size,
from the [population in the context of the problem], 95% of the intervals constructed
using this method would capture the true [population parameter from context of the
problem].
The Distinction between Confidence Interval and Confidence Level
A confidence interval is a range of plausible values contained in an interval that we
construct by applying a particular statistical method to data from a single sample. We are
confident that this range of plausible values contains the population parameter in
question.
The confidence level refers to the width of the interval, not the interval itself. Statistical
confidence is expressed as a percentage. This percentage refers to the percentage of
intervals across repeated samples, created using a particular statistical method, that
capture the true population parameter.