1 9.1 Hypothesis Testing (part 2) Introduction to Hypothesis Testing Null hypothesis (H 0 )? Or the Alternative hypothesis (H a )?
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9.1 Hypothesis Testing (part 2)
Introduction to Hypothesis
Testing
Null hypothesis (H0)?
Or the Alternative hypothesis (Ha)?
Who makes a claim or a statement?
A company: The amount of cereal advertised on a box.
An everyday person: One person claims she is a more accurate
basketball shooter than another person. A researcher:
They claim that their new drug is better than a presently used drug on the market.
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How do we decide if the claim is believable? Or unbelievable?
A company: Measure the amount in a sample of boxes.
An everyday person: Record the number of baskets made and missed.
A researcher: Assign some patients to each of the drugs and
see how the patients fare on the drugs. 3
We collect data!!!!
How do we decide if the claim is believable? Or unbelievable?
… and then we do a . What is a ?
It is a standard procedure for testing a claim about the value of a population parameter.
It is a way to make a decision based on evidence (collected data).
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We collect data!!!!
The Procedure
We will have two “competing” hypotheses:
The null hypothesis This statement always has an = sign in it. The mean amount of cereal in the box is 16 oz. H0: µ = 16 oz.
The alternative hypothesis This is where the ‘counter statement’ goes This statement has one of these in it: <, > , ≠ HA: µ ≠ 16 oz.
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The Procedure
Claim: The average cost of a veterinary clinic visit is greater than $100.
The null hypothesis H0: µ = $100
The alternative hypothesis HA: µ > $100
6 NOTE: Cost is a continuous variable parameter of interest is µ.
The Procedure
Claim: The proportion of students who like Friday afternoon classes is less than 0.10.
The null hypothesis H0: p = 0.10
The alternative hypothesis HA: p < 0.10
7 NOTE: We’re collecting ‘like’ or ‘not like’ responses parameter of interest is p.
The Procedure Claim: A sales rep claims that her vending
machines dispense coffee so that the mean amount supplied is equal to 10 ounces.
The null hypothesis H0: µ = 10 ounces
The alternative hypothesis HA: µ ≠ 10 ounces
8 NOTE: Ounces is a continuous variable parameter of interest is µ.
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Data in a sample is used to calculate a statistic, such as (sample mean) or s (sample std. dev.). Values that describe the population are parameters, such as µ (population mean) or p (population proportion).
X
The Procedure
A hypothesis states a claim about a parameter (µ or p), not a statistic.
But we use a calculated statistic, such as to make a decision about a population parameter, such as µ.
is our best estimate for µ, so it makes sense to use that in our decision about µ.
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X
X
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An Example… Claim: An engine company claims their
engine has a mean octane rating of 90.
The null hypothesis H0: µ = 90
The alternative hypothesis HA: µ ≠ 90
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Example 1: Test of Hypothesis for
Step 1: State your null and alternative hypotheses.
µ
90:H90:H0
≠
=
µ
µ
A
We use ≠ because ‘greater than’ and ‘less than’ were not part of the original statement.
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Step 2: Draw a sample of engines and calculate (sample mean) and s (sample standard deviation).
x
Example 1: Test of Hypothesis for µ
n = 60 randomly sampled engines
= 90.94 s = 2.45
x
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Step 3: Construct the 95% confidence interval for µ.
Formula:
Calculated:
Example 1: Test of Hypothesis for µ
X ± 2sn
90.94± 2(2.45)60
(90.307, 91.573)
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Step 3 (continued): Construct the 95% confidence interval for µ.
We are 95% confident that the true population mean octane rating for all engines built by this company is between 90.307 and 91.573.
Example 1: Test of Hypothesis for µ
(90.307, 91.573)
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Step 4: Use the 95% confidence interval to make a decision for the hypothesis test.
95% CI: 90 is NOT in the CI, so 90 is NOT a plausible
value for the population mean octane rating.
Example 1: Test of Hypothesis for µ
90:H90:H0
≠
=
µ
µ
A
(90.307, 91.573)
Decision Reject H0.
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There is convincing evidence that the population mean octane rating is not 90 but something different.
Example 1: Test of Hypothesis for µ
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The Procedure Because we have used a 95% confidence
interval to perform this hypothesis test, the significance level of the test is at the 0.05 level.
In this class, we will always perform tests at the 0.05 level, but other levels are possible.
For example, a test done using a 90% confidence interval would be at the 0.10 level.
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Example 2: Test of Hypothesis for µ
I think the population mean number of songs in an iTunes library is 7000.
The null hypothesis H0: µ = 7000
The alternative hypothesis HA: µ ≠ 7000
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Step 1: State your null and alternative hypotheses. H0 :µ = 7000
HA :µ ≠ 7000
Example 2: Test of Hypothesis for µ
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Step 2: Draw sample and calculate the statistics and s.
x
= 7160 songs, s=1200, from n=150 sample x
Example 2: Test of Hypothesis for µ
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Step 3: Construct the 95% confidence interval for µ.
Formula:
Calculated:
Example 2: Test of Hypothesis for µ
X ± 2sn
7160± 2(1200)150
(6964, 7356)
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Step 4: Use the 95% confidence interval to make a decision for the hypothesis test.
95% CI: 7000 IS in the CI, so 7000 IS a plausible value
for the population mean number of songs.
Example 2: Test of Hypothesis for µ
Decision Do Not Reject H0.
H0 :µ = 7000HA :µ ≠ 7000
(6964, 7356)
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There is not convincing evidence that the population mean number of songs is something other than 7000.
Example 2: Test of Hypothesis for µ
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The Procedure
We NEVER prove that the null is correct.
If we do not reject the null (i.e. we accept it), it just means we didn’t have strong evidence for the alternative hypothesis.
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The Procedure The hypothesis test procedure starts off by
assuming that the null is true, and then we collect evidence (or data).
The evidence tells us one of two things... 1) We should ‘throw-out’ the null statement
that we originally assumed to be true because the data support the alternative hypothesis.
2) We do not have strong evidence to support the alternative hypothesis (kind of wishy washy outcome).
27 27 Copyright © 2009 Pearson Education, Inc.
1. Reject the null hypothesis, H0, in which case we have evidence in support of the alternative hypothesis.
2. Do not reject the null hypothesis, H0, in which case we do not have enough evidence to support the alternative hypothesis.
Two Possible Outcomes of a Hypothesis Test