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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:

Equation 1

    1. The standard normal distribution is:

Equation 2

The normal standard distribution

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

    2. The observation is standardized by

Equation 3

  1. Form a confidence interval

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

Equation 4

    1. If a = 5%, then

A normal standard distribution with alpha = 5%

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

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

Equation 5

      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

Equation 1

    1. Take a random sample

    2. n = 90, The mean = 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)

Equation 6

    1. Form a 95% Confidence Interval

Equation 7

A 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

The z and t distributions

    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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