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CHAPTER 8.2
CHAPTER 8 ESTIMATION
PART 2 – Estimating  When  Is Unknown
ADVANCED PROBABILITY AND STATISTICS:
 16.0 – Students know basic facts concerning the relation between the mean
and the standard deviation of a sampling distribution and the mean and
the standard deviation of the population distribution.
 17.0 – Students determine confidence intervals for a simple random sample
from a normal distribution of data and determine the sample size
required for a desired margin of error.
OBJECTIVE(S):
 Students will learn the definition of a Student’s t distribution, degrees
of freedom.
 Students will learn how to calculate the margin of error.
 Students will learn how to find the critical value knowing a confidence
level.
 Students will learn how to estimate the mean  when  is unknown.
 Students will learn which distribution you should use when estimating
the mean  .
Student’s t Distribution
In order to use the normal distribution to find confidence intervals for a population
mean  , we need to know the value of _____, the population __________________
___________________________. Typically when _____ is unknown, _____ is
unknown as well. When we use ____ to approximate _____, the sampling distribution
for _____ follows a new distribution called a ____________________ ____
________________________.
Assume that x has a normal distribution with mean _____. For samples of size n with
sample mean x and sample standard deviation s, the t variable
t = _____________
has a Student’s t distribution with degrees of freedom d.f. = ____________.
Properties of a Student’s t distribution
1.) The distribution is symmetric about the mean 0.
2.) The distribution depends on the degrees of freedom.
3.) The distribution is bell-shaped, but has thicker tails than the standard normal
distribution.
CHAPTER 8.2
4.) As the degrees of freedom increases, the t distribution approaches the standard
normal distribution.
Area Under the t Curve Between _____ and ______
Area = c
 tc
0
tc
P (______ < t < _______) = ____
1.) Find the critical value tc for a 0.99 confidence level for a t distribution with sample
size n = 5.
STEP 1: Find the column with c heading ___________.
STEP 2: Compute the number of degrees of freedom:
d.f. = ___________= ______________ = ______.
STEP 3: Read down the under the heading c = ________ until we reach the row
headed by ____ (under d.f.).
The entry is _______________. Therefore, t 0.99 = ________.
If the degrees of freedom d.f. you need are not in the table, use the closest d.f. in the table
that is _________________. This procedure results in a critical value tc that is more
conservative in the sense that it is larger. The resulting confidence interval will be longer
and have a probability that is slightly higher than ____.
CHAPTER 8.2
2.) Find the critical value tc for a 0.90 confidence level for a t distribution with sample
size n = 9.
STEP 1: Find the column with c heading ___________.
STEP 2: Compute the number of degrees of freedom:
d.f. = ___________= ______________ = ______.
STEP 3: Read down the under the heading c = ________ until we reach the row
headed by ____ (under d.f.).
The entry is _______________. Therefore, t 0.90 = ________.
3.) Find the critical value tc for a 0.95 confidence level for a t distribution with sample
size n = 9.
Maximal Margin of Error, E
E=
Confidence Interval for  with  Unknown
A c confidence interval for  is an interval computed from sample data in such a way
that c is the probability of generating an interval containing the actual value of  .
P (__________ < ____ < ___________) = __
How to find a confidence interval for  with  unknown
Let x be a random variable appropriate to your application. Obtain a simple random
sample (of size n, n>1) of x values from which you compute the sample mean x and the
sample standard deviation s.
If you can assume that x has a normal distribution or simply a mound-shaped symmetric
distribution, then any sample size n will work. If you cannot assume this, then use a
sample size of n  30 .
Confidence Interval for  with  Unknown
________________ <  < ______________
where x = sample mean of a simple random sample
CHAPTER 8.2
E=
___________
c = ________________ ________________ (___< c < ____)
tc = critical value for confidence level c and degrees of freedom d.f. = _________
4.) Suppose an archaeologist discovers only seven fossil skeletons from a previously
unknown species of miniature horse. Reconstruction of the skeletons of these seven
miniature horses show the shoulder heights (in centimeters) to be
45.3 47.1 44.2 46.8 46.5 45.5 47.6
The mean is x  _______ and the sample standard deviation is s  _______. Let
 be the mean shoulder height (in centimeters) for this entire species of miniature
horse, and assume that the population of shoulder heights is approximately
normal.
Find a 99% confidence interval for  , the mean shoulder height of the entire
population of such horses.
n = ______
d.f. = _________ = ___________ = ______
c = ______
therefore, t 0.99  ______________
E
=
=

tc
s
n
_______________
CHAPTER 8.2
The 99% confidence interval is
xE
<

<
<

<
<

<
xE
The archaeologist can be ______ confident that the interval from ___________ to
_____________ is an interval that contains the population mean  for shoulder
height of this species of miniature horse.
CALCULATOR
Press the STAT key and select TESTS, then use 7:TInterval. Select Stats and
then below inputs, x , n, and C-Level. Highlight Calculate and hit enter.
Output:
TInterval
(_______________, ______________)
x = _________
s = ______
n = ______
5.) A company has a new process for manufacturing large artificial sapphires. In a trial
run, 37 sapphires are produced. The mean weight for these 37 gems is x =6.75
carats, and the sample standard deviation is s = 0.33 carat. Let  be the mean weight
for the distribution of all sapphires produced by the new process.
a. What is d.f. for this setting?
b. Find E.
CHAPTER 8.2
c. Find a 95% confidence interval for  .
d. Interpret the confidence interval in the context of the problem.
Which distribution should you use for x
Examine problem statement
 is known
Use normal
distribution with
margin of error
E  zc

n
 is unknown
Use Student’s t
distribution with
margin of error
E  tc
s
n
d.f. = n - 1