Fundamentals of hypothesis testing: one-sample tests

Question 1

Hypothesis

Answer 1

A hypothesis is a statement (assumption) about a population parameter

Question 2

Population mean example

Answer 2

Example: The mean monthly mobile phone bill of this city is μ = $72

Question 3

Population proportion example

Answer 3

Example: The proportion of adults in this city with mobile phones is ∏ = 0.89

Question 4

The Null Hypothesis, H0

Answer 4

States the belief or assumption in the current situation (status quo)

Begin with the assumption that the null hypothesis is true
(similar to the notion of innocent until proven guilty)

Refers to the status quo

Always contains ‘=‘, ‘≤’ or ‘’ sign

May or may not be rejected

Is always about a population parameter; e.g. μ, not about a sample statistic

Question 5

The Alternative Hypothesis, H1

Answer 5

Is the opposite of the null hypothesis
e.g. The average number of TV sets in Australia
homes is not equal to 3 ( H1: μ ≠ 3 )

Challenges the status quo

Can only can contain either the ‘’ or ‘≠’ sign

May or may not be proven

Is generally the claim or hypothesis that the researcher is trying to prove

Question 6

Hypothesis testing process

Answer 6

PHOTOS 1-2

Question 7

The Level of Significance, alpha

Answer 7

hypothesis is true
Defines rejection region of the sampling distribution

Is designated by alpha, (level of significance)
Typical values are 0.01, 0.05, or 0.10
Note relationship to 99%, 95% and 90% confidence levels

Is selected by the researcher at the beginning

Provides the critical value(s) of the test

Question 8

Level of Significance and the Rejection Region

Answer 8

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Question 9

Errors in making decisions

Answer 9

Type I error
Reject a true null hypothesis
Considered a serious type of error

Type II error
Fail to reject a false null hypothesis

Question 10

The probability of errors

Answer 10

The probability of Type I error is alpha
Called level of significance of the test; i.e. 0.01, 0.05, 0.10
Set by the researcher in advance

The probability of Type II error is β

Question 11

Outcomes and probabilities of hypothesis testing

Answer 11

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Question 12

Z Test of Hypothesis for the Mean (σ Known)

Answer 12

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Question 13

Critical Value Approach to Testing

Answer 13

For a two-tail test for the mean, σ known:
Convert sample statistic ( ) to the test statistic (Z statistic)
Determine the critical Z values for a specified level of significance  from a Table E.2 or computer
Decision Rule: If the test statistic falls in the rejection region, reject H0 ; otherwise do not reject H0

Question 14

Two tail tests

Answer 14

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Question 15

6 STEPS IN HYPOTHESIS TESTING

Answer 15

State the null hypothesis, H0 and the alternative hypothesis, H1

Choose the level of significance, alpha, and the sample size, n

Determine the appropriate test statistic and sampling distribution

Determine the critical values that divide the rejection and non-rejection regions

Collect data and calculate the value of the test statistic

Make the statistical decision and state the managerial conclusion

Question 16

Step 6 expanded

Answer 16

if the test statistic falls into the non-rejection region, do not reject the null hypothesis H0; if the test statistic falls into the rejection region, reject the null hypothesis

express the managerial conclusion in the context of the real-world problem

Question 17

Hypothesis testing example

Answer 17

photos 7-10

Question 18

p-value approach to testing

Answer 18

p-value: Probability of obtaining a test statistic more extreme
( ≤ or ) than the observed sample value, given H0 is true

Also called observed level of significance
Smallest value of  for which H0 can be rejected
Obtain the p-value from Table E.2 or computer
If p-value < alpha , reject H0
If p-value >= alpha , do not reject H0

Question 19

p value example

Answer 19

Photos 11-12

Question 20

Connection to confidence intervals

Answer 20

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Question 21

One tails tests

Answer 21

Photo 14

Question 22

Lower tail tests

Answer 22

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Question 23

Upper tail tests

Answer 23

Photo 16

Question 24

Upper tail test example

Answer 24

Photos 17-20

Question 25

Upper tail test p-value example

Answer 25

Photo 21

Question 26

t test of hypothesis for the mean (sigma unknown)

Answer 26

Photo 22

Question 27

t test of hypothesis for the mean (sigma unknown) example

Answer 27

Photos 23-24

Question 28

Hypothesis Tests for Proportions

Answer 28

Photos 25-26

Question 29

Hypothesis Tests for Proportions example

Answer 29

Photos 27-28

Question 30

The power of a test (1 – β)

Answer 30

Photo 29

Question 31

Type II Error

Answer 31

Photos 30-31

Question 32

Pitfalls and Ethical Considerations

Answer 32

Use randomly collected data to reduce selection biases or coverage errors

Do not use human subjects without informed consent

Choose the level of significance, α, and the type of test (one-tail or two-tail) before data collection

Do not employ ‘data snooping’ to choose between one-tail and two-tail test, or to determine the level of significance

Do not practice ‘data cleansing’ to hide observations that do not support a stated hypothesis

Report all pertinent findings

Distinguish between statistical significance vs. practical significance