1. Confidence intervals for m when s is unknown
When s is unknown, we use the sample standard deviation s to estimate s. In place of z-scores, we must use the following to standardize the values:
The use of the value of s introduces extra variability. Therefore the distribution of t values has more variability than a standard normal curve.

2. Important Properties of t Distributions
The t distribution is bell shaped and centered at zero (just like the standard normal (z) distribution).
Each t distribution is more spread out than the standard normal distribution.
z curve
t curve for 2 df

3. 3) The shape of a particular t-distribution curve depends on df.

4. The number of degree of freedom is defined as the number of observation that can be chosen freely.
df = n-1
Example-1:
Find the value of t for 16 df and 0.05 area in the right tail of a t-distribution curve.

6. 4) As the number of degrees of freedom increases, the spread of the corresponding t distribution decreases.
t curve for 8 df
t curve for 2 df

7. 5) As the number of degrees of freedom increases, the corresponding sequence of t distributions approaches the standard normal distribution.
For what df would the t distribution be approximately the same as a standard normal distribution?
z curve
t curve for 2 df
t curve for 5 df

8. How to find t-critical using GDC?
Menu
Statistics (6)
Distributions (5)
Inverse t (6)
Note that Area here is “Area to the left of the t-critical value”

9. Examples to Try
1. Find the value of t for a t-distribution with 14 degree of freedom and 0.01 area in the upper tail.
2. Find the value of t for a t-distribution for confidence level 90% and n=19

10. Confidence intervals for m when s is unknown
The general formula for a confidence interval for a population mean m based on a sample of size n when . . .
1) x is the sample mean from a random sample,
2) the population distribution is normal, or the sample size n is large (n > 30), and
3) s, the population standard deviation, is unknown is
Where the t critical value is based on df = n - 1.
This confidence interval is appropriate for small n ONLY when the population distribution is (at least approximately) normal.

11. Example-2
Ten randomly selected shut-ins were each asked to list how many hours of television they watched per week. The results are
Find a 90% confidence interval estimate for the true mean number of hours of television watched per week by shut-ins.

12. Calculating the sample mean and standard deviation we have n = 10, = 86, s = 11.842
We find the critical t value of by looking on the t table in the row corresponding to df = 9, in the column with bottom label 90%. Computing the confidence interval for  is

13. To calculate the confidence interval, we had to make the assumption that the distribution of weekly viewing times was normally distributed. Consider the normal plot of the 10 data points produced with Minitab that is given on the next slide.

14. p-value is more than 0.05 we assume that the distribution is normal
Notice that the normal plot looks reasonably linear so it is reasonable to assume that the number of hours of television watched per week by shut-ins is normally distributed.
Typically if the
p-value is more than 0.05 we assume that the distribution is normal
P-Value:
A-Squared: 0.226
Anderson-Darling Normality Test

15. How to use GDC to find the intervals????????
to be taught on Wednesday!

16. Test on ch-9 (to be discussed and decided) Thursday 21st, January 2016 or Monday 22nd, Feb 2016 or Tuesday 23rd , Feb 2016 ??