|PART FOUR Essentials| -- The Normal Distribution -- The red section may be omitted
Normal distribution characteristics--center concentration and symmetry
Parameters of the normal distribution--μ (mu), mean; and σ (sigma), standard deviation
Z-value formula (four arrangements--for z, x, μ, and σ)
Normal distribution problems have three variables given, and the fourth must be
computed and interpreted
Z-values determine areas (probabilities) and areas (probabilities) determine z-values--the normal
table converts from one to the other
Normal distribution probability tables--our text table presents one-sided central areas
Two uses of the normal distribution
Normally-distributed phenomena
To approximate the binomial distribution--this application is far less important now that computers
and even small calculators can generate binomial probabilities
Binomial parameters (n and p) can be converted to normal parameters μ and σ
μ = np; σ2 = (npq); σ = (npq)
Continuity correction--needed when a continuous distribution (e.g. normal) is used to
Approximate a discrete (e.g. binomial) distribution
Three causes for the normal-binomial discrepancy
Normal is continuous; binomial is discrete
Normal is infinite; binomial is finite
Normal is always symmetrical; binomial is symmetrical only when p = q
Conditions for good normal-binomial approximation: large n; p near q
Poisson distribution a better approximator of the binomial when p is not near q
Poisson is discrete, like the binomial
Poisson is finite, like the binomial
Poisson is skewed, like the binomial when p and q are not equal
Sampling Distributions
Sampling distribution of the mean--the distribution of the means of many samples of the same size
Drawn from the same population
Central Limit Theorem--three statements about the sampling distribution of the means:
1. Sampling distribution of the means is normal in shape, regardless of the population distribution
shape when the sample size, n, is large. (When n is small, the population must be normal in
order for the sampling distribution of the mean to be normal.) ("Large" n is usually taken
to mean 30 or more.)
2. Sampling distribution of the means is centered at the true population mean.
3. Sampling distribution of the means has a standard deviation equal to σ / n.
This quantity is called the sampling standard deviation or the standard error (of the mean).
(The full name is "standard deviation of the sampling distribution.”)
This quantity is represented by the symbol σxbar.
σxbar is less than σ because of the offsetting that occurs within the sample. The larger
the sample size n, the smaller the σxbar (standard error), because the larger the
n, the greater the amount of offsetting that can occur, and the sample means will
cluster more closely around the true population mean μ.
Sampling standard deviation (σxbar or standard error)--key value for inferential statistics
Two uses of the standard error
Computing the error factor in interval estimation
Computing the test statistic (zc or tc) in hypothesis testing
Terminology--explain each of the following:
normal distribution, normal distribution parameters, mean, standard deviation, standard normal
distribution, z-value, reliability, validity, sampling distribution, central limit theorem (three parts),
sampling standard deviation, standard error, offsetting, effect of the sample size on the sampling
standard deviation (standard error).
Skills and Procedures--given appropriate data,
 determine a normal probability (area), given x, μ, and σ
 determine x, given μ, σ, and the normal probability (area)
 determine μ, given x, σ, and the normal probability (area)
 determine σ, given x, μ, and the normal probability (area)
 solve various applications problems involving the normal distribution
 compute the sampling standard deviation (standard error) from the population standard
deviation and the sample size
 solve various applications problems involving the central limit theorem
Concepts- describe conditions under which the normal distribution is symmetric
 describe the kind of shift in the graph of a normal distribution caused by a change in the
mean
 describe the kind of shift in the graph of a normal distribution caused by a change in the
standard deviation
 explain why, as the sample size increases, the distribution of sample means clusters more
and more closely around the population mean