1. WARM - UP 1. How do you interpret a Confidence Interval?
We are C% confident (sure) that the true proportion/average of the population for this study is between x1 and x2.
2. How do you interpret a Confidence Level?
If we obtained many samples of this similar size, we can expect that C% of them will produce an interval that captures the true average of the population for this study.
3. What is the z* for a 95%Confidence Level?
z95* = This is called a critical value

2. Chapter 23 t – DISTRIBUTIONS (Means)
When we are unrealistically given the known population standard deviation ‘σ’, we use the standard normal distribution = z – test statistic:
BUT…The Std. Dev, σ is usually unknown and has to be estimated. This estimate is called the Standard Error =
Inferences based on unknown σ (usually samples less than n=30) are made with the t–distribution (t – test statistic)
There is a different t-distribution for each sample size P-values are computed with ‘degrees of freedom’.
Degree of Freedom: df = n – 1 (Sample Size – 1)

3. Finding Probability from t :
The Test Statistic for a Test of the Hypothesis: H0: μ = μ0 with an unknown population σ is computed by:
The P-Value is then computed by:
Ha: μ > μ0 P( T ≥ t0 ) tcdf(t, E99, df)
Ha: μ < μ0 P( T ≤ t0 ) tcdf(-E99, t, df)
Ha: μ ≠ μ0 2P( T ≥ |t0| ) 2·tcdf(|t|, E99, df)
P-Val.
= t

4. The P-Value is then computed by:
Ha: μ > μ0 P( T ≥ t0 ) tcdf(t, E99, df)
Ha: μ < μ0 P( T ≤ t0 ) tcdf(-E99, t, df)
Ha: μ ≠ μ0 2P( T ≥ |t0| ) 2·tcdf(|t|, E99, df)
EXAMPLES: Use Calculator 2nd Vars #6 tcdf
Find the P-Value for t ≥ 2.58 with 12 degrees of freedom.
Find the P-Value for | t | > 1.62 with 20 degrees of freedom.
3. Find the P-Value for t ≤ 0.67 with a sample size of 4.
tcdf(2.58, E99, 12) =
P-Val =
2∙tcdf(1.62, E99, 20) =
P-Val =
tcdf(-E99, 0.67, 3) =
P-Val = 0.725

5. Computing the critical value - t* from Probability:
TABLE B: Find the intersection of the Probability (to the Right) on the top row or Confidence Level on the bottom row with the appropriate degree of freedom.
CALCULATOR: Use 2nd Vars #4
INV T (Probability to the Left, df)
EXAMPLES:
1. Find the critical value t* for a t statistic from a SRS of 28 observation with probability of .80 to the left.
2. Find the critical value t* for a t statistic from a SRS of 38 observation with probability of .025 to the right.
INV T(.8,27) t = 0.855
t = 2.021
INV T(.975,37) t = 2.026

6. Computing the critical value - t
Computing the critical value - t* from Confidence Levels: Find the probablity to the left by:
CALCULATOR: Use 2nd Vars #4 INV T (Probability to the left, df)
Make the Answer Positive.
EXAMPLES:
1. Find the critical value t* for a 95% confidence interval with a sample size of 15.
2. Find the critical value t* for a 90% confidence interval with a sample size of 9.
INV T(.025,14) t = 2.145
INV T(.05,8) t = 1.860

7. HW: Page 541: #1,2, 5,6,9

8. HW: Page Page 541: #1,2, 5,6,9