Statements of Probability and Confidence Intervals
Lecture 4

The z Distribution and Confidence Intervals

1. If data is normally distributed, we can calculate a standard normal distribution

   1. A normal distribution is:

2. The standard normal distribution is:

3. If m = 68, s ² = 100, and the 87 ^th observation is X ₈₇ = 70

4. The observation is standardized by

2. Form a confidence interval

   1. Usually set a = 5% (or 0.05). It is okay to have an a = 10% or a = 1%

2. If a = 5%, then

3. For two sided confidence intervals, we usually put a/2 in each tail

4. Thus, z _{a /2} = z _0.025 = 1.96 for a standard normal

5. Example

   1. = 68, which is an unbiased estimate for the population parameter, m

   2. The standard deviation is s = 10 and a = 0.05

3. We would expect 95% of the data to fall between [48.4, 87.6]

3. Standard Errors – use one sample to determine variability of population parameter, m

   1. We have the following distribution

2. Take a random sample

3. n = 90, = 110, and s ² = 81

4. We are assuming we know the variance now; usually this is unknown too!

5. We calculate the standard error (SE)

6. Form a 95% Confidence Interval

7. There is a 95% chance that the true population mean lies between [108.1, 111.9]

4. We assume we know s ²

   1. However, we have to estimate s ² too

   2. We switch the distribution to a t-distribution

3. The t-distribution is shorter with fatter tails

   1. Uses degrees of freedom

   2. df = n – 1

   3. The one is we estimated the variance, so we lose one piece of information

   4. As the degrees of freedom approaches infinity, the t-distribution collapses onto the normal distribution

   5. As the sample size becomes larger, the standard error becomes smaller. The confidence intervals become smaller too!