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Elementary Statistics

- Chapter 1

The Scientific Method

- Step 1 Formulate a Theory.
- Step 2 Collect data to test the theory.
- Step 3 Analyze the results.
- Step 4 Interpret the results and make a

decision.

Definitions

- Population - the entire group of objects or

individuals under study. - Sample - a part of the population that is

actually used to get information. - Statistical inference - the process of drawing

conclusions about the population based on

information from a sample of that population.

Statistical Hypothesis

- A statistical hypothesis consists of two parts,

the null hypothesis and the alternate hypothesis.

Statistical Hypothesis

- Null Hypothesis - denoted by H0 - the

conventional belief - the status quo, or

prevailing viewpoint, about a population.

- Alternate Hypothesis - denoted by H1 - an

alternate to the null hypothesis - the change in

the population that the researcher hopes to find

evidence for.

Statistically Significant

- The data collected are said to be statistically

significant if they are very unlikely to be

observed under the assumption that H0 is true.

If the data are statistically significant, then

our decision would be to reject H0.

What if we make a mistake?

- Type I Error Rejecting the null hypothesis when

in fact it is true.

- Type II Error Accepting the null hypothesis

when in fact it is not true.

Decisions, Decisions

Definition

- The Significance Level number ? is the chance of

committing a Type I error -- that is the chance

of rejecting the null hypothesis when it is in

fact true.

Frequency (or Dot) Plot

- A frequency plot displays a set of observations

by representing each observation value with a

symbol positioned along a horizontal scale. If

there are two or more observations with the same

value, the symbols are stacked vertically.

Definition

- The number n of observations in a sample is

called the sample size.

Decision Rule

- A decision rule is a formal rule that states,

based on the data obtained, when to reject the

null hypothesis H0. Generally, it specifies a

set of values based on the data to be collected,

which are contradictory to the null hypothesis H0

and which favor the alternative hypothesis H1.

Direction of Extreme

- The Direction of extreme corresponds to the

position of the values that are more likely under

the alternative hypothesis H1 than under the null

hypothesis H0. If the larger values are more

likely under H1 than under H0, then the direction

of extreme is said to be to the right.

Most Extreme Value

- The value under the null hypothesis H0 that is

least likely, but at the same time is very likely

under the alternative hypothesis H1, is called

the most extreme value.

Rejection Region

- A rejection region is the set of values for which

you would reject the null hypothesis H0. Such

values are contradictory to the null hypothesis

and favor the alternative hypothesis.

Rejection Region

- An acceptance region is the set of values for

which you would accept the null hypothesis H0.

- A cutoff value, or critical value, is a value

that marks the starting point of a set of values

that comprise the rejection region.

One or Two - Sided?

- A rejection region is called one-sided if its set

of extreme values are all in one direction,

either all to the right or all to the left.

- A rejection region is called two-sided if its set

of extreme values are in two directions, both to

the right and to the left.

P-value

- The p-value is the chance, computed under the

assumption that H0 is true, of getting the

observed value plus the chance of getting all of

the more extreme values.

Two Approaches for Decision Making

- Classical Approach Choose a level of

significance, ?, and use it to create a decision

rule.

- P-value Approach Collect the data and calculate

the likelihood of the data based on the

assumption that the null hypothesis is true. If

the p-value is less than ?, reject H0.

Think About It

- The significance level is ? 0.1, the chance of

committing a Type I error. The corresponding

decision rule is Reject H0 if the selected

voucher is 50 or more. A voucher is selected

and it turns out to be 60. Your decision is to

reject the null hypothesis and conclude that the

data are statistically significant at the 10

level.

- You reject H0. Could you have made a mistake?

- What type of mistake could you have made?

- What is the chance that you have made a mistake?

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