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STAM4000

Quantitative Methods

Week 8

Hypothesis testing

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3 ing #1 #2 #3 Describe the steps of hypothesis testing Construct hypothesis tests for one population mean Examine errors in hypothesis testing |
Week 8 Hypothesis test Learning Outcomes |

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Why does this matter?

We sometimes need to

determine if there is

significant evidence to

support a claim.

(http://4.bp.blogspot.com/-zf7S5L0XT-U/T_PX-wWXEBI/AAAAAAAADp8/DPKrX_iJJUA/s1600/1b.jpg)

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#1 Describe the steps of hypothesis testing

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#1 What is a hypothesis?

A hypothesis is an

idea,

claim or

belief

about a population,

that we want to test,

using a sample.

Photo by Jonathan Daniels on Unsplash

We are continuing with INFERENTIAL STATISTICS,

where we use a sample to infer or draw conclusions

on the population.

Learn the steps in hypothesis testing

Step 1: Write the hypotheses

Step 2: Find the calculated test statistic and/or the p-value

Step 3: Find the critical value

Step 4: Sketch the curve, showing the rejection region

Step 5: Decision

Step 6: Conclusion

Notes:

• Step 1, hypotheses must be stated in advance, before sampling etc.

• Steps 2, 3 and 4 may be interchanged

• Steps 5 and 6 must follow this order.

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#1 Steps in hypothesis testing

Test for μ, so we

need symbol of μ or

u in our

hypotheses.

Use a formula to

COMPARE the x

bar with u from

our hypotheses.

Critical value is

from the

“statistical

tables”

Sketch a

curve for

perspective

or context

Comparison,

e.g.: compare

critical value with

calculated value

Ties the decision to

the original

question

Hypotheses

• Null hypothesis, Ho

o Read as “H nought”

o Expresses what we initially ASSUME to be true.

• Alternative Hypothesis, denoted by Ha, HA, or H1

o Expresses our claim into a statement that we are trying to gather

enough evidence to PROVE is NOW true.

The goal in hypothesis testing is to determine whether there is enough evidence to

infer (or draw the conclusion) that the null hypothesis is no longer true and the

alternative hypothesis is now true.

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#1 Step 1: Write the hypotheses

•Expresses what we initially ASSUME to be true. |

•Expresses our claim into a statement that we are trying to gather enough evidence to PROVE is NOW true. |

Null hypothesis, denoted by

Ho

Alternative Hypothesis,

denoted by

Ha, HA, or H1

Ho is read as “H nought”

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An individual is thought to have committed a crime and is brought before a court

of law.

In Australia, we presume the individual is innocent,

then gather evidence to try and prove they are guilty.

We have two hypotheses:

Ho: individual is innocent, ASSUME THIS IS TRUE

Ha: individual is guilty, PROVE THIS IS NOW TRUE

Objective of

testing:

gather evidence

to reject Ho and

accept Ha.

#1 Introductory example

Introductory Example

An individual is caught committing a crime.

This individual is prosecuted in a court of law.

It is assumed the individual is innocent.

Evidence is gathered to try to reject innocence and conclude the individual is, in fact, guilty.

How do we represent this in hypotheses?

We write two hypotheses:

Ho: individual is innocent

Ha: individual is guilty

The goal in hypothesis testing is to determine whether there is enough evidence to infer (or

draw the conclusion) that the alternative hypothesis is true.

(https://www.youtube.com/watch?v=tug71xZL7yc)

(https://www.reddit.com/r/aww/comments/2y40ts/lawyer_cat_is_on_retainer/)

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10 | Two Correct outcomes |
Two Incorrect outcomes (errors) |

Possible outcomes in our introductory example?

Innocent

individual is freed

Guilty individual is

imprisoned

Innocent

individual is

imprisoned

Guilty individual

is freed

Four Possible outcomes

#1

Possible outcomes in our introductory example?

Hypotheses,

Ho: individual is innocent

Ha: individual is guilty

Four possible outcomes:

Two correct outcomes:

i) innocent individual is freed

ii) guilty individual is imprisoned

Two incorrect outcomes (errors)

i) innocent individual is imprisoned

ii) guilty individual is freed

Later, we will discuss errors in testing, in detail.

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Quick quiz

This Photo by Unknown Author is licensed under CC BY

Let’s use the previous example, with the following

hypotheses:

Ho: individual is innocent, ASSUME Ho TRUE

Ha: individual is guilty, Prove Ho is false, and Ha is TRUE

Which is worse:

• an innocent individual going to prison

or

• a guilty individual going free?

#1

An innocent individual going

to prison is WORSE.

This type of error occurs if

we INCORRECTLY REJECT Ho.

We have INCORRECTLY

accepted Ha of guilt.

This type of error is called a

TYPE I error.

P(TYPE I ERROR) = α, the

level of significance.

Quick Quiz

Let’s use the previous example, with the following hypotheses:

Ho: individual is innocent

Ha: individual is guilty

Which is worse:

• An innocent individual going to prison

or

• a guilty individual going free?

The worst of these is an innocent individual going to prison.

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The calculated test statistic is a value calculated using a formula, which has been

specifically chosen, to compare the population scenario in the null hypothesis and the

values from a sample.

The p-value is the corresponding probability-value of the calculated test statistic.

If we are looking at a sketch of a curve:

• The calculated test statistic(s) would lie on the horizontal axis.

• The p-value is the area(s) under the curve, corresponding to the calculated test

statistic value(s) , or more extreme) in the tail(s), if Ho were really true.

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#1 Step 2:Find the calculated test statistic and/or the p-value

•Value from a formula, quantifying the difference between what is the population and what is in the sample. |

•The probability of getting our sample results or more extreme, if our null hypothesis were really true. |

hypothesised in

Calculated test statistic

p-value : is a probability

value related to the

calculated test statistic

and the direction in Ha.

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p-value has become

more popular with

technology easily

providing p-values.

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The critical value is that value which determines the rejection region, hence is critical

in the decision step of the hypothesis test process.

The critical value, in part, relies on the level of significance, α.

Some properties about α:

• Read as “alpha”.

• Is also called the significance level.

• The level of significance, α, is the area or probability in the tail(s) of a curve.

• α and the direction in Ha help us define the rejection region by finding the critical

value(s).

• Usually, 1% ≤ α ≤ 10% or 0.01 ≤ α ≤ 0.10.

• α should be selected by the researcher before a sample is selected. Why? The

selection of α, has an impact on the rejection region and hence on the decision.

• α is the probability of incorrectly rejecting Ho. This is P(Type I error) – see later.

• α is the probability associated with the critical value.

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#1 Step 3: Find the critical value

•As a minimum, we need: o Relevant statistical tables or technology o Level of significance, α, or SIGNIFICANCE LEVEL o Number of tails in Ha |

Critical value

•Area of rejection region or rejection regions

•Probability of Type I error (later)

Level of significance, α

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Ho:

Ha:

Objective is to reject Ho and accept Ha, usually the case.

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Sketch a curve:

The type of curve we sketch, depends on the distribution we are using for the

hypothesis test e.g.’ Z or t – more later

The rejection region(s), will lie in the tail(s) of the curve.

For the purpose of a hypothesis test, we can divide the curve into two main sections:

• Horizontal axis: this follows the scale of the variable.

• Area beneath the curve: this is the probability related to the calculated test

statistic value and the direction of the alternative hypothesis

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#1 Step 4: Sketch a curve

• oOn axis of curve: |

▪Insert critical value(s), axis for TODAY ▪Label rejection region(s) = AREA UNDER CURVE THAT TOTALS TO ▪Insert calculated value, from the formula oIn area beneath curve ▪Insert α ▪Insert p-value (if relevant), is the |

Ha determines the number of tails in a curve

on the Z or t

α

probability related to the calculated

test statistic AND direction in Ha

Sketch a curve: for

perspective or context,

for yourself

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Decision:

There are two possible methods we can use in the decision step of hypothesis testing:

i. Critical value method: comparing values on the horizontal axis of the curve

or

ii. p-value method: comparing areas (probabilities) beneath the curve

Whether we choose the critical value method or the p-value method depends on the information we

are given/question asked. The p-value method has become more popular with the progress of

technology.

The decision should include:

• Whether the null hypothesis is rejected or retained.

• The process to make that decision, in the context of the question i.e. the comparison of the

calculated test statistic with the critical value OR the comparison of the p-value with α.

• The significance level, α, used to make that decision. Why? Different significance levels, will have

different critical values and rejection regions which may/may not result in different decisions and

conclusions.

• In research, usually the sample size is also noted.

Conclusion:

This is the most important step in hypothesis testing.

Here, we are tying our decision to the original question – i.e. we are answering the original question.

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#1 Steps 5: Decision and Step 6: Conclusion

i. Critical value method: compare calculated test statistic with critical value(s). or ii. p-value method: compare p-value, of calculated test statistic, with α |

Decision

If we reject Ho, then we may accept Ha.

BUT, if we fail to reject Ho, then we must RETAIN Ho;

we NEVER accept Ho.

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Conclusion •Use your decision to answer the original question.

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Critical value method:

compares VALUES ON

THE AXIS OF THE

CURVE

p-value method:

compares AREAS

under the curve.

These two methods

will give us the SAME

DECISION, but they are

focused on two

different aspects of

the curve.

Whether we

choose to use the

critical method or

the p-value

method:

• information

provided

• wording of the

question

Ho: innocent

Ha: guilty

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#2 Construct hypothesis tests for one population mean, μ

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Test hypotheses about one population mean, μ

In this class, we will be using quantitative data and testing for values of the population mean, μ.

• Both our null and alternative hypotheses will include the population mean symbol, μ.

• The null hypothesis will contain an “=“ symbol.

• The alternative hypothesis will have a direction determined by the wording of the question.

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#2 Test hypotheses about one population mean, μ

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• Read as “alpha”.

• α is also called the significance level.

• α, and the direction in Ha, will help us define the rejection region by finding the critical

value(s).

• Usually, 1% ≤ α ≤ 10% or 0.01 ≤ α ≤ 0.10.

• α of 1% is a very strict test, as it has a very small rejection region, area totalling 0.01

• α of 10% is a more lenient test, as it has a larger rejection region, area totalling 0.10

• α should be selected by the researcher before a sample is selected.

• α is the probability of incorrectly rejecting Ho.

#2 The level of significance, α

How is α used in confidence intervals, as compared to hypothesis testing?

In Confidence intervals:

• The confidence level is (1 -α)

• The confidence interval % is (1 -α)%

• With confidence intervals, we are interested in a “central range” of the curve.

In hypothesis testing:

• The tail(s) of the curve have a total area (probability) of α.

• We are interested in the tail(s) of the curve.

Can a confidence interval be used for a hypothesis test? Yes. Its straightforward for a twotailed test for the population mean, μ,when the population standard deviation, σ, is known.

If the null hypothesis value of the population mean, μ, lies outside in the CI, we reject Ho.

Otherwise, we retain Ho.

Note: we will concentrate on the hypothesis test steps to test hypotheses.

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α

= P(Type I error)

= probability of incorrectly rejecting Ho

We may choose the level of α

we are willing to tolerate for a hypothesis test.

In your assessment, α

will be specified:

• α

• level of significance

• significance level

in either % or decimal form.

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Denotes

position of

critical

Ho: μ = value value(s)

Ha : μ < value, left-tailed, directional 0

Ho: μ = value One tailed test in the right or upper tail |
a |

Ha : μ > value, right-tailed, directional

a

One tailed test in the left or lower tail

0

Two-tail test REJECTION REGION |
/2 REJECT Ho |

Ho: μ = value | |

Total area = 100% or 1 | REJECT Ho |

Rejection

region(s) is

shaded

0

a/2 | a | Do rejec |
ot t Ho |

Ha: μ ≠ value, non-directional

#2 Writing hypotheses and sketches to test μ

α is the

level of

significance

REJECTION REGION

Write the hypotheses to test μ: to test for values of the population mean,