﻿ Statistical Methods - Lecture 4 HcWjnyVHiTd8hN_8STvJ2rWaXvhPz4wXYCNGvD4qDkU
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# 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:

1. The standard normal distribution is:

1. If m = 68, s2 = 100, and the 87th observation is X87 = 70

2. The observation is standardized by

1. Form a confidence interval

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

1. If a = 5%, then

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

2. Thus, za/2 = z0.025 = 1.96 for a standard normal

3. Example

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

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

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

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

1. We have the following distribution

1. Take a random sample

2. n = 90, = 110, and s2 = 81

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

4. We calculate the standard error (SE)

1. Form a 95% Confidence Interval

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

1. We assume we know s2

1. However, we have to estimate s2 too

2. We switch the distribution to a t-distribution

1. 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!

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