1. For One Sample Test Around the Mean.
1.1 Calculating Confidence
Intervals
A Confidence Interval is an interval or range of numbers.
It is calculated around a "point estimate" (such as the mean of a
group of numbers). We calculate confidence intervals for continuous numbers because
we can not estimate one number out of infinitely. We use confidence
intervals to estimate how far the point estimate is from the parameter of a
population of a specific level of confidence.
Large Sample Confidence Intervals for a Population Mean
Z Statistic – A statistic having the standard normal
(or Gaussian) distribution. It arises in test of a mean which is compared to a
known variance.

Large Sample > or = to 30

Use when population variances are known or unknown

Sampling From Independent Normal Distributions

One Sided or Two Sided test

One mean/average known
The following formulas are for two sided test.
Confidence intervals come with two sides.
You may also use the t test formula if population variance
is unknown.
Small Sample Confidence Intervals for a Population Mean
Degrees of Freedom – Are the number of independent elements
that go into a statistic. An independent comparison. Number of pieces of
information that are independent of one another in that they cannot be deduced
from one another. The t test requires one to use Degrees of Freedom with
looking up the correct values in the t tables.
When the level of confidence and sample standard deviation remain the same, a
confidence interval for a population mean based on a sample of n = 100 will be
narrower than a confidence interval for a population mean based on a sample of n
= 50.
Information
created by Craig Stevens Based on material found in a couple of different
texts.
Bottom
more appropriate for confidence intervals (two sided in nature).
Determine the Sample Size
B = the value farthest away from the population that you will
allow the sample mean to be. the "n" will be the size of the
sample required for us to be confident (to some degree, i.e. 95%) of our
answer.
Confidence Intervals for Parameters of Finite Populations
 point estimate = a onenumber estimate of the value of a population
parameter
 population parameter = a descriptive measure of the population)
 p = population proportion
 p hat = sample proportion
 When the sample size and the sample proportion remain the same, a 90%
confidence interval for a population proportion (p hat) will be narrower
than the 99% confidence interval for p. The bell shaped curve will
have narrower distance between the standard deviations.
 When the level of confidence and sample proportion remain the same, a
confidence interval for a population proportion (p hat) based on a sample
of n = 100 will be wider than a confidence interval for p based on a
sample of n = 400
 When the level of confidence and sample size remain the same, a
confidence interval for a population proportion p will be wider when is
larger than when it is smaller.
Determining Sample Size for Confidence Interval for p
B = the value farthest away from the population that you will
allow the sample mean to be. the "n" will be the size of the
sample required for us to be confident (to some degree, i.e. 95%) of our
answer.
p = population proportion
Confidence Intervals for Population Mean and Total for a Finite
Population
For a large (n > or = to 30) random sample of measurements selected
without replacement from a population of size N.
Confidence Intervals for Proportion of and Total Number of Units
when Sampling a Finite Population
For a large random sample of measurements selected without replacement from
a population of size N
1.2 Hypothesis Testing
Null and Alternative Hypotheses
Ho = The null hypothesis, is a statement of the basic proposition being
tested. The statement generally represents the status quo and is not rejected
unless there is convincing sample evidence that it is false.
Ha = The alternative or research hypothesis, is an alternative (to the null
hypothesis) statement that will be accepted only if there is convincing sample
evidence that it is true.
Six Steps of Hypothesis Testing
 Determine null and alternative hypotheses
 Specify level of significance (probability of Type I error) ?
 Select the test statistic that will be used.
 Collect the sample data and compute the value of the test
statistic.
 Use the value of the test statistic to make a decision using a rejection
point or a pvalue.
 Interpret statistical result in (realworld) managerial terms
Examples From our Textbook
Large Sample Tests about a Mean: Testing a OneSided Alternative
Hypothesis
If the sampled population is normal or if n is
large, we can reject H0:
m
= m0 at
the a
level of significance (probability of Type I error equal to a)
if and only if the appropriate rejection point condition holds.
Large Sample Tests about Mean: pValues
The
pvalue = the observed level of significance = the
probability of observing a value of the test statistic greater than or equal to
z when Ho is
true.
It measures the weight of the evidence against the null hypothesis and is
also the smallest value of alpha
for which we can reject Ho.
Large Sample Tests about a Mean: Testing a TwoSided Alternative
Hypothesis
Small Sample Tests about a Population Mean
Hypothesis Tests about a Population Proportion
Type II Error Probabilities and Sample Size Determination
The ChiSquare Distribution
Statistical Inference for a Population Variance
