1. Confidence Intervals for  With  Unknown

2. CONFIDENCE INTERVALS FOR  WHEN  IS UNKOWN
In real life, we rarely know 
 IS UNKNOWN!
In situations where value of  is unknown, then if we can assume:
samples come from a normal distribution (i.e. X has a normal distribution)
The random variable has what is called a
t-distribution

3. The t-distribution X is normal and  is unknown
The t-distribution is “robust” which means that it can be used even if X is only approximately normal
X is normal and
 is unknown

4. SHAPE OF THE t-DISTRIBUTION
The t-distribution is a mound-shaped distribution just like the z distribution, only more spread out because of the added uncertainty about 
The larger the sample size, the less unsure we are about  and thus the t-distribution approaches the z distribution as the sample size, n, gets large

5. DEGREES OF FREEDOM
A measure of how spread out the t-distribution is provided a quantity known as the number of degrees of freedom
The number of degrees of freedom is a measure of how the sample size affects estimating the standard deviation of the population

6. DEGREES OF FREEDOM WHEN ESTIMATING 
In hypothesis testing and confidence intervals for the mean, , we calculate
Thus the degrees of freedom for a sample of size n is n-1

7. t-DISTRIBUTIONS

8. t TABLES
For each possible number of degrees of freedom, there exists a table for t that is just like the normal table
This is too bulky and so typically one t-table is printed that gives the t-values for common probabilities (in the tail): t.10, t.05, t.025, t.01, and t.005 for degrees of freedom up to 30, and for other selected number of degrees of freedom

9. t-Distribution With 9 df
The t-Table
t.01,9
2.821
.01
t-Distribution With 9 df t
t-tables give right tail probabilities

10. Properties of t-Distributions
Mean = 0
Symmetric
Example: P(t < ) = P(t > 2.821)
Example: (t-value) that puts .01 in the right tail = (-t-value) that puts .01 in the left tail
The larger the degrees of freedom, the shape of the t-distribution becomes closer to a normal distribution
t-values are sometimes approximated by z-values if n > 30
As will be shown, there is no need to do this with EXCEL

11. CONFIDENCE INTERVALS WHEN  IS UNKNOWN
Exactly the same as when using z except:
Use t instead of z
Use s instead of the unknown 

12. Confidence Intervals for 
When  is known we used:
When  is unknown we use:
Note: we include
the # of
Degrees of Freedom

13. EXAMPLE
Assuming that the ages of MIS managers follow a normal distribution, estimate the true mean age of MIS managers with 95% confidence given the following 5 observations: 25, 30, 32, 38, 25

14. 95% CONFIDENCE INTERVAL FOR 

15. t Functions in EXCEL
TINV(.05,6) = the t value that puts .025 in the upper tail with 6 degrees of freedom
TDIST(1.783,7,1) = area in the upper tail above with 7 degrees of freedom
TDIST(1.783,7,2) = twice the area in the upper tail above with 7 degrees of freedom
NOTES:
The first argument of TDIST must be positive.
tα/2 is returned when TINV(α,DF) is entered.
Since any number of DF can be entered, there is no need to approximate t by z when using Excel.

16. Manipulating t-Functions in Excel
The probability to the left of t = with 7 degrees of freedom: =1 - TDIST(1.783,7,1)
The probability in the lower tail below t = with 7 degrees of freedom: = TDIST(1.783,7,1)
The probability to the right of t = with 7 degrees of freedom: =1 - TDIST(1.783,7,1)
The probability between and with 7 degrees of freedom: = 1 – TDIST(1.783,7,2)

17. EXCEL CONFIDENCE INTERVALS
Go to DESCRPTIVE STATISTICS -- Check
Summary Statistics
Confidence Level for Mean (indicate % confidence)
LCL = (Mean)- (Confidence)
UCL = (Mean)+(Confidence)
Numbers in italics means click on the cell with this value

18. Confidence Level For Mean
CHECK --
Summary statistics
Confidence Level For Mean
Degree of
Confidence

19. = D3-D16
= D3+D16

20. REVIEW t distribution is used when
Sampling from (relatively) normal distributions
 is unknown
t distribution is based on degrees of freedom (DF)
DF for tests and intervals about  = n-1
t-intervals are the same as z-intervals except
use s instead of 
use t instead of z
Excel –
TDIST and TINV
Descriptive Statistics to Create Confidence Intervals