Download Section 7-2: Confidence Intervals for the Mean When σ Is Unknown

Section 7-2: Confidence Intervals for the Mean
When σ Is Unknown
When one of the following is true, the confidence interval
for the mean can be found by using the standard normal
(Z) distribution in Section 7-1:
  is known and the population is normally distributed
or
 the sample size is at least 30.

When  is unknown, we can no longer use the standard
normal distribution, but instead we will use a t
distribution.

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Characteristics of the t-distribution
The t distribution is similar to the standard normal
distribution in these ways:
 It is bell-shaped
 It is symmetric about the mean
 The mean, median, and mode are equal to 0 and are
located at the center of the distribution
 The curve never touches the x-axis.
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The t distribution differs from the standard normal
distribution in the following ways:
 The variance is greater than 1 (i.e., the t distribution has
more variability than a standard normal distribution).
 The t distribution is actually a family of curves (see pg.
786) based on the concept of degrees of freedom, which is
related to sample size.
 The shape of the t curve changes depending on the
degrees of freedom.
 As the sample size increases, the t distribution becomes
more and more liked the standard normal distribution.
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Degrees of freedom (df) - the number of values that are free
to vary after a sample statistic has been computed.
Example: Suppose the mean of 5 values if 10. What is  x ?
How many values can vary freely?
We could vary 4 values of the 5.

The degrees of freedom for a confidence interval for the mean
when  is unknown is found by
Sample size – 1 or n-1
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Formula for a Specific Confidence Interval for the Mean
When σ is Unknown
The formula for a confidence interval about the mean when 
is unknown is given by
 s 
X t 2  ,
 n 

where t /2 is the 1  /2 quantile of a t distribution with n 1
degrees of freedom.


We willuse Table F to find our t multiplier, t /2
.
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
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ASSUMPTIONS:
1. Our sample is a random sample from
some population.
2. The population from which the
sample came is normally distributed
or the sample size n30.
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Example: Find the t multiplier (t  2) for 98% confidence and 24
degrees of freedom.
1-=0.98 so we need =0.02  t/2 = 2.492

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Example: Find the t-multiplier (t  2) for 90% confidence and
17 degrees of freedom.

Note: Why do we need to use the t distribution instead of the Z
distribution? Since we are estimating the population standard
deviation  , we need to construct a wider confidence interval
to take into account the uncertainty in our estimate of  . The t
distribution changes the multiplier to the proper number to get
the correct confidence level.


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Example: A sample of six college wrestlers had an average weight of
276 pounds with a sample standard deviation of 12 pounds. Find
the 95% confidence interval of the true mean weight of all college
wrestlers. If a coach claimed that the average weight of the
wrestlers on his team was 310, would the claim be believable?
Assumptions: We have a random sample of college wrestlers. Since
our sample size is under 30, we must assume that the average
weight is normally distributed.
X  276
s  12
n  6 t /2  2.571 df  6-1 = 5
 s 
 12 
X  t /2 

276

2.571


  276  12.59528  (263, 289)
 n
 6




Interpretation: We are 95% confident that the average weight of a
college wrestler is between 263lbs and 289lbs.
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Example: The daily salaries of substitute teachers for eight local
school districts is shown. What is the sample mean? Find the 99%
confidence interval of the mean for the salaries of substitute teachers
in the region.
60 56 60 55 70 55 60 55
Assumptions:
X
s
 s 
X t /2  
 n 


n
t /2 

df 

Interpretation:
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Constructing confidence intervals on a TI-83/84 for the Mean
When σ is Unknown:
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Entering data on a TI-83/84:
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When do you use the z-table and when do you use the t-table?!?!
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SOME EXTRA CALCULATOR
HELP…DON’T FORGET OUR
YOUTUBE PLAYLIST.
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