Introduction to the normal distribution and the standard normal which is in units of Z (see p. 341) where Z =

X - m

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standard deviation

Introduces you to the central limit theorem. The theorem is really about how samples from a population behave. If you could look at a graph of all possible samples, they would be normally distributed around the parameter with a spread called the STANDARD ERROR.

For sample means, the population parameter being estimated is m and the population standard deviation is s so the standard error is s/÷ n

For sample proportions, the population parameter being estimated is p and the standard error is ÷ [p*(1-p)]/n

Both of these theoretical formulas are given on page 451.

You are interested in buying a business. The owner claims his customers spend an average (m) of $1,200 with a standard deviation (s) of $1000. He allows you to draw a random sample (n) of 150 customers from his database.

Suppose your sample of 150 customers has an average (x-bar) of $950 and a standard deviation (S) of $200.

If the manufacturer is truthful, how likely is it to get a sample average of $950 or less?

The standard deviation of the sampling distribution is 1000/ (root (150)) = 81.6497 dollars,

The chance of getting an average of $950 or less

is Z = (950-1200)/81.6497 = -3.06

about 0.11% or like something like 1 in a 1000 samples

NOTE: we are able to calculate a CHANCE OR PROBABILITY HERE. For example -- 0.11%

Sometimes (most times) you are in a situation where you do not know the value of the parameters or you only know one of them. In those situations, we cannot calculate a chance or a probability but we make STATEMENTS OF CONFIDENCE or construct CONFIDENCE INTERVALS.

A confidence interval is basically a statement of belief. Verbally, one would say

"I am 'C' percent confident that the true parameter is covered by the interval

x-bar ± Za _/2(s / ÷ n) (for situations where you are trying to estimate a population mean)

Where Za _/2 can be read from your standard normal table. C = (100% - a ) so if you were 68% confident, then alpha = 32% and the multiplier Za _/2 would correspond to two tail areas of 16% each. A Z = 1.00 and -1.00 would satisfy this.

If you wanted to be 90% confident, then alpha = 10% and the multiplier Za _/2 would correspond to two tail areas of 5% each. A Z = 1.645 and -1.645 would satisfy this.

FOR PROPORTIONS (11.6) it only differs a little.

"I am 'C' percent confident that the true parameter is covered by the interval

p-hat ± Za _/2(÷ [p-hat*(1-p-hat)]/n) (for situations where you are trying to estimate a population proportion)

Where Za _/2 can be read from your standard normal table. C = (100% - a ) so if you were 68% confident, then alpha = 32% and the multiplier Za _/2 would correspond to two tail areas of 16% each. A Z = 1.00 and -1.00 would satisfy this.

If you wanted to be 90% confident, then alpha = 10% and the multiplier Za _/2 would correspond to two tail areas of 5% each. A Z = 1.645 and -1.645 would satisfy this.