Chapter 9: Inference for Means
9.1 A Confidence Interval for a Mean
Objectives
 To construct a confidence interval for a mean using the sample standard deviation and tdistribution.
 To interpret a confidence interval for a mean.
 To compute the margin of error.
The method for constructing a confidence interval for a mean is the same as that used in chapter 8,
however there is one important difference. Different distributions are used depending on whether  is
known or not.
Confidence z-Interval for a Mean (standard deviation known)
Although it is not realistic that the population standard deviation is known, we will look briefly at how to
find a confidence interval for a sample where it is known.
This CI is based on the Central Limit Theorem, which says that when n is sufficiently large (generally
 30), the sampling distribution of x is approximately normal for any population distribution.
æ s ö
x ± ( z critical value )× ç
è n ÷ø
Conditions:
1.
2.
3.
Individuals are independent. The sample is a
simple random sample (or random assignment of
treatments to units).
In the case of a sample survey, the population
size should be at least ten times as large as the
sample size.
Population is Normal. The underlying
population is normal (histogram is unimodal and
symmetric) or the sample is large (n  30).
If n is small, but it is reasonable to believe that the distribution of values in the population is normal, the
same CI formula can still be used.
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9.1 A Confidence Interval for a Mean
Example: A plant produces steel sheets whose weights are known to be normally distributed with a
standard deviation of 2.4 kg. A random sample of 36 sheets had a mean weight of 31.4 kg. Find the 99%
CI for the population mean.
Student t Distribution
A Student t distribution is another type of probability distribution. Normal distributions are distinguished
from one another by their mean and standard deviation; however, t distributions are distinguished by a
positive whole number called the number of degrees of freedom (df). There is a t distribution with 1 df,
another with 2 df, and so on.
Important Properties of t Distributions






The t distribution is bell shaped and centered at zero (just like the standard normal distribution).
The t distribution is different for different sample sizes.
Each t distribution is more spread out (flatter) than the standard normal distribution.
As the number of degrees of freedom increases the spread of the corresponding t distribution
decreases.
As the number of degrees of freedom increases, the corresponding sequence of t distributions
approaches the standard normal distribution. For n > 30 the differences are negligible.
The population is essentially normal.
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9.1 A Confidence Interval for a Mean
To use a t distribution, the first step is to calculate a t-score (much like calculating a z-score for the
normal distribution): t =
x -m
s
n
Next you calculate the degrees of freedom (df) = n – 1. In general, the degrees of freedom is the number
of values that can vary after certain restrictions have been imposed on all values. Suppose, for example,
there were 10 tests that averaged 55. If you assign nine people random grades, the last test score is not
random, but constrained by the overall mean. Thus for 10 tests and a mean, there are nine degrees of
freedom.
Finally you look a probability up on a table or on your calculator.
Confidence t-Interval for a Mean (standard deviation unknown)
Realistically the standard deviation is not known. Because you are going to use s to approximate  , you
must use a t distribution, instead of a normal distribution.
 s 
x  (t critical value )  

 n
Conditions:
1.
2.
3.
3
Individuals are independent. The sample is a
simple random sample (or random assignment of
treatments to units).
In the case of a sample survey, the population
size should be at least ten times as large as the
sample size.
Population is Normal. The underlying
population is normal (histogram is unimodal and
symmetric) or the sample is large (n  30).
9.1 A Confidence Interval for a Mean
The extra uncertainty that results from estimating  causes the t interval to be wider than the z interval.
Example: The mass, in grams, of a packet of biscuits of a particular brand, follows a normal distribution
with mean  . Ten packets of biscuits are chosen at random and their masses noted. The results, in grams,
are:
397.3, 399.6, 401.0, 392.9, 396.8, 400.0, 397.6, 392.1, 400.8, 400.6
Calculate a 90% confidence interval for the mean.
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9.1 A Confidence Interval for a Mean