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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

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#1 Steps in hypothesis testing

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#1 Step 1: Write the hypotheses

•Expresses what we initially ASSUME to be true. |

Null hypothesis, denoted by

Ho

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

Alternative Hypothesis,

denoted by

Ha, HA, or H1

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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

Ha: individual is guilty

Objective of

testing:

gather evidence

to reject Ho and

accept Ha.

#1 Introductory example

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10 | 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

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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

Ha: individual is guilty

Which is worse:

• an innocent individual going to prison

or

• a guilty individual going free?

#1

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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 hypothesised the population and what is in the sample. |

Calculated test statistic

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

p-value

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#1 Step 3: Find the critical value

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

Critical value

•Area of rejection region •Probability of Type I error (later) |

Level of significance, α

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#1 Step 4: Sketch a curve

• oOn axis of curve: ▪Insert critical value(s) ▪Label rejection region(s) ▪Insert calculated value oIn area beneath curve ▪Insert α ▪Insert p-value (if relevant) |

Ha determines the number of tails in a curve

Sketch a curve

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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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#2 Construct hypothesis tests for one population mean

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#2 Test hypotheses about one population mean

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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

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

• 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

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Denotes

position of

critical

value(s)

Ho: μ = value

Ha : μ < value 0

Ho: μ = value

Ha : μ > value

a

a

One tailed test in the left or lower tail

0

One tailed test in the right or upper tail

Two-tail test

Rejection

region(s) is

shaded

/2

0

Ho: μ = value a

Ha: μ ≠ value

Total area = 100% or 1

#2 Writing hypotheses and sketches to test μ

α is the

level of

significance

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#2 Example

i) The manager of a soda drink

company has set the bottling

machine to fill bottles to an

average volume of 250 ml.

An assembly line worker claims

the bottling machine is not

filling the bottles correctly.

The worker samples 40 bottles

and finds an average of 230 ml

per bottle.

Write the hypotheses to test the

worker’s claim.

ii) A phone

company

executive announced that the

customer call waiting time, for

their helpline, is less than, an

average of 12 minutes per call.

The company takes a random

sample of 36 calls and finds an

average wait time of 11 minutes.

Write the hypotheses to test the

executive’s announcement.

iii) The CEO of a fast-food

franchise reported that the

average weekly sales, per

franchise, is $45,000. The

marketing team believes this

can be increased and runs an

advertising campaign. The CEO

sampled the weekly sales of 10

franchises, after the campaign,

and found average sales were

$47,000.

Write the hypotheses to test

the marketing team’s belief.

This Photo by Unknown Author is licensed under CC BY-SA

This Photo by Unknown Author is licensed under

CC BY

This Photo by Unknown Author is licensed

under CC BY-NC-ND

Ho: µ = 250

Ha: µ ≠ 250

Ho: µ = 12

Ha: μ < 12

Ho : µ = 45,000

Ha: µ > 45,000

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#2

Test statistic

formula to test μ

σ is KNOWN

use Z