1. 7.2 ESTIMATING μ
WHEN σ IS UNKNOWN
Chapter 7: Estimation

2. Page
347
 Usually, when  is unknown,  is unknown as
well. In such cases, we use the sample
standard deviation s to approximate .
 A Student's t distribution is used to obtain
information from samples of populations with
unknown standard deviation.
A Student’s t distribution depends on sample size.

3. Page
348
Student’s t Distribution
 The variable t is defined as follows:
Assume that x has a normal distribution with
mean μ. For samples of size n with sample
mean and sample standard deviation s, the t
variable
has a Student’s t distribution with degrees
of freedom d.f. = n – 1
 Each choice for d.f. gives a different t distribution.

4. 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.
4. As the degrees of freedom increase, the t
distribution approaches the standard normal
distribution.
5. The area under the entire curve is 1.
Figure 7-5
A Standard Normal Distribution and
Student’s t Distribution with
d.f. = 3 and d.f. = 5

5. Page
349
Finding Critical Values
 Table 6 of Appendix II gives various t values for
different degrees of freedom d.f. We will use this
table to find critical values tc for a c confidence
level.
 In other words, we want to find tc such that an
area equal to c under the t distribution for a
given number of degrees of freedom falls
between –tc and tc in the language of
probability, we want to find tc such that
P(–tc  t  tc) = c
Figure 7-6
Area Under the t Curve Between –tc and tc

6. Finding Critical Values:
Using Table 6
1. Find the column with the c heading
2. Compute the degrees of freedom and find the
row that contains the d.f.
3. Match the column and row
Convention for using Student’s t distribution
If the d.f. you need are not in the table, use the closest
d.f. in the table that is smaller.

7. Page
349
Example 4 – Student’s t Distribution
Find the critical value tc for a 0.99 confidence
level for a t distribution with sample size n = 5.
Student’s t Distribution Critical Values (Excerpt from Table 6, Appendix II)
Table 7-3
t0.99 = 4.604

8. Page
350 Confidence Interval for μ when σ is
Unknown
 Requirements
Let x be a random variable appropriate to your application. Obtain a
simple random sample (of size n) of x values from which you compute
the sample mean and the sample standard deviation s.
If you can assume that x has a normal distribution or is mound-
shaped, then any sample size n will work.
If you cannot assume this, then use a sample size of n ≥ 30.
 Confidence Interval for μ when σ is unknown
where
= sample mean of a simple random sample
d.f. = n – 1
= confidence level (0 < c < 1)
= critical value

9. Not in Textbook!
How To Construct a Confidence
Interval
1. Check Requirements
 Simple random sample?
 Assumption of normality?
 Sample size?
 Sample mean?
 Sample standard deviation s?
2. Compute E
3. Construct the interval using

10. Page
Example 5 – Confidence
351
Interval


11. Solution – Confidence Interval

The archaeologist can be 99% confident that
the interval from 44.5 cm to 47.8 cm is an
interval that contains the population mean  for
shoulder height of this species of miniature
horse.

12. Using the Calculator
1. Hit STAT, tab over TESTS, Choose 8:Tinterval
2. Highlight STATS, hit ENTER
3. Enter the requested information
4. Highlight Calculate, Hit Enter
Note: The solution will be listed in the format
(lower value, upper value)

13. Page
353
Summary: Which Distribution
Should You Use?

14. Assignment
 Page 354
 #1, 4 – 7, 11 – 15 odd