1. Section 8.2 Estimating  When  is Unknown
In order to use the normal distribution to find the confidence intervals for a population mean μ, we need to know the value of σ, the population standard deviation. However much of the time, when μ is unknown, σ is unknown as well. In such cases, we use the sample standard deviation s to approximate σ. When we use s to approximate σ, the sampling distribution for follows a new distribution called a Student’s t distribution.

2. Student’s t Distributions
The student’s t distribution uses a variable t defined as follows. A student’s t distribution depends on sample size n. Assume that x has a normal distribution with mean μ. For samples of size n with sample mean and standard deviation s, the t variable has a Student’s t distribution with degree of freedom d.f. = n – 1.

3. The shape of the t distribution depends only on the sample size, n, if the basic variable x has a normal distribution.
When using the t distribution, we will assume that the x distribution is normal.
Appendix Table 4 (Page A8) gives values of the variable t corresponding to the number of degrees of freedom (d.f.)
d.f. = n – 1
where n = sample size

4. The t Distribution has a Shape Similar to that of the the Normal Distribution

5. 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 (d.f. = b-1 for μ confidence intervals.
3. The distribution is Bell-shaped with thicker tails than the standard normal distribution.
4. As the degrees of freedom increase, the t distribution approaches the standard normal distribution

6. Using Table 4 to find Critical Values for confidence Intervals
Table 4 of the Appendix gives various t values for different degrees of freedom d.f. We will use the table to find the critical values for a c confidence level. In other words, we want to find such that the area equal to c under the t distribution for a given number of degrees of freedom falls between and
In the language of probability, we want to find such that
This probability corresponds to the shaded area in next figure

7. Using Table 4 to Find Critical Values tc for a c Confidence Level

8. Using Table 4 to Find Critical Values of tc
Find the column in the table with the given c heading
Compute the number of degrees of freedom: d.f. = n  1
Read down the column under the appropriate c value until we reach the row headed by the appropriate d.f.
If the required d.f. are not in the table:
Use the closest d.f. that is smaller than the needed d.f.
This results in a larger critical value tc.
The resulting confidence interval will be longer and have a probability slightly higher than c.

9. Example: Find the critical value tc for a 95% confidence level for a t distribution with sample size n = 8.
Find the column in the table with c heading 0.950

10. Example cont. Compute the number of degrees of freedom:
d.f. = n  1 = 8  1 = 7
Read down the column under the appropriate c value until we reach the row headed by d.f. = 7

11. Example cont.
tc = 2.365

12. PROCEDURE How to find Confidence Intervals for  When  is Unknown
c = confidence level (0 < c < 1)
tc = critical value for confidence level c, and degrees of freedom d.f. = n  1

13. Confidence Intervals for ,  is Unknown
Example
Confidence Intervals for ,  is Unknown
The mean weight of eight fish caught in a local lake is 15.7 ounces with a standard deviation of 2.3 ounces. Construct a 90% confidence interval for the mean weight of the population of fish in the lake.
Solution
Mean = 15.7 ounces Standard deviation = 2.3 ounces.
n = 8, so d.f. = n – 1 = 7
For c = 0.90, Appendix Table 4 gives t0.90 =

14. The 90% confidence interval is:
Example cont.
E = 1.54
The 90% confidence interval is:
<  <
14.16 <  < 17.24
We are 90% sure that the true mean weight of the fish in the lake is between and ounces.

15. Summary: Confidence intervals for the mean
Assume that you have a random sample of size n > 1 from an x distribution and that you have computed and c. A confidence interval for μ is Where E is the margin of error. How do you find E? It depends of how of how much you know about the x distribution.

16. Summary cont.
Situation 1 (most common) You don’t know the population standard deviation σ. In this situation, you use the t distribution with margin of error and degree of freedom d.f. = n – 1 Although a t distribution can be used in many situations, you need to observe some guidelines. If n is less than 30, x should have a distribution that is mound-shaped and approximately symmetric. Its even better if the x distribution is normal. If n is 30 or more, the central limit theorem implies that these restrictions can be relaxed

17. Summary cont.
Situation 2 (almost never happens!) You actually know the population value σ. In addition, you know that x has a normal distribution. If you don’t know that the x distribution is normal, then your sample size n must be 30 or larger. In this situation, you use the standard normal z distribution with margin of error

18. Summary cont. Which distribution should you use for
Examine problem statement
If σ is known
use normal distribution with margin of error
If σ is not known
use Student’s t distribution with margin of error
d.f. = n - 1
Assignment 20