ACMS10145 Lecture Notes - Lecture 9: Null Hypothesis, Statistical Hypothesis Testing, Broccoli

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19 Feb 2019
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Chapter 9: Hypothesis Tests
We are constantly bombarded by quantitative claims in the media. Advertisers make claims
(e.g., 9 out of 10 dentists recommend Crest toothpaste), political candidates make claims
(e.g. “most American do/do not want universal healthcare”), and scientists make claims (e.g.
eating broccoli will reduce your risk of cancer by x %), and so on. Statistical hypothesis
testing provides a method for us to test these claims quantitatively and objectively.
9.1 Developing Null and Alternative Hypotheses
We wish to evaluate the statistical evidence to decide between two competing hypotheses,
the Null Hypothesis (H0) and the Alternative Hypothesis (Ha):
H0: Tentative assumption about the population parameter.
Ha: A statement that is opposed of what is stated in the null hypothesis H0.
Situations when identifying hypothesis:
a. The Alternative Hypothesis as Research Hypothesis.
Example:
b. The Null Hypothesis as an assumption to be challenged.
Example:
1Anew drug comes out and is claimed to be
better than the existing ones
2Anew fuel injection system is designed
to increase the miles per gallon rating
consider the situation of amanufacturer
of asoft drink product The label of the bottle
states it contains 500 mL
Ho µ7500 mL
Ha µL500 mL
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Summary of forms
H0
Ha
There are two possible decisions: ____________________ ___________________
Steps of Hypothesis Testing
1. Develop the Null and Alternative Hypotheses
2. Specify the level of significance, !.
3. Collect the sample data and compute the value of the test statistic
p-value Approach:
4. Use the test statistic to compute the p-value.
5. Reject H0 if p-value !; otherwise, fail to reject H0.
Critical Value Approach:
4. Use the level of significance to determine critical value and rejection rule.
5. Compare the test statistic to the critical value to decide whether to reject H0.
onetailed upper µEno µNo
onefailed lower µ7pro te CMo
Two failed test µpeo µ4Mo
Reject Ho Fail to reject Ho
Ho Ha
0what data
Conman L01 05 1tells us
O
Compute 22 Za Hh Hh Fa ta th tag
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9.2 Type I and Type II Errors
There are two types of error we can make when we decide whether to accept the Null
Hypothesis or reject the Null hypothesis. The concept of a rejection region leads us to the
concept of type 1 error.
TYPE I Error = Reject H0 when H0 is true.
The Level of Significance of the test, denoted by ! is the probability that we make a type
I error.
Eg:
! = The probability that we reject H0 when H0 is in fact true.
TYPE II Error = Fail to reject H0 when H0 is false.
We let # denote the probability that we make a type II error. The value of # is usually
unknown, because the alternative hypothesis is usually less specific than the null
hypothesis.
Eg:
# = the probability that we fail to reject the null hypothesis when the null hypothesis is in
fact false.
Taste Testing as a Hypothesis Test
Who can tell the difference between Coke and Diet Coke? Can you prove it? We will
randomize the contents of n=10 cups by flipping a coin.
H0: Ha:
We let the test statistic be X, the # of correct guesses.
Under H0, what is the distribution of X?
Ffathereis no fire
there is fire
The firealarm goes off whenthere
is no fire PType Ierror
The firealarm doesn't go off when there is give
PC
TypeTt errot Test result
peality Reject Ho Fail to rejectHo
Ho true Type Ierror
Ho false Type Il error
He she can not tell Heshe can distinguish
the difference bwCoheDiet
cohe coldDiet Coke
Xhas binomial distribute'm n10 p5
X0I9,10
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Document Summary

We are constantly bombarded by quantitative claims in the media. Statistical hypothesis testing provides a method for us to test these claims quantitatively and objectively. We wish to evaluate the statistical evidence to decide between two competing hypotheses, the null hypothesis (h0) and the alternative hypothesis (ha): Ha: a statement that is opposed of what is stated in the null hypothesis h0. Situations when identifying hypothesis: the alternative hypothesis as research hypothesis. better than. 1 a new drug comes out and is claimed to be. 0: develop the null and alternative hypotheses, specify the level of significance, !. Conman: collect the sample data and compute the value of the test statistic p-value approach, use the test statistic to compute the p-value, reject h0 if p-value ! Critical value approach: use the level of significance to determine critical value and rejection rule, compare the test statistic to the critical value to decide whether to reject h0.

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