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Section 8.2Basics of Hypothesis Testing

Objective

For a population parameter (p, µ, σ) we wish to test whether a predicted value is close to the actual value (based on sample values).

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DefinitionsIn statistics, a Hypothesis is a claim or statement about a property of a population.

A Hypothesis Test is a standard procedure for testing a claim about a property of a population.

Ch. 8 will cover hypothesis tests about a• Proportion p• Mean µ (σ known or σ unknown)

• Standard Deviation σ

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Claim: The XSORT method of gender selection increases the likelihood of birthing a girl.

(i.e. increases the proportion of girls born)

To test the claim, use a hypothesis test (about

a proportion) on a sample of 14 couples:If 6 or 7 have girls, the method probably doesn’t increase the probability of birthing a girl.

If 13 or 14 couples have girls, this method probably does increase the probability of birthing a girl.

This will be explained in Section 8.3

Example 1

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Rare Event Rule forInferential Statistics

If, under a given assumption, the probability of a particular event is

exceptionally small, we conclude the assumption is probably not correct.

Example:

Suppose we assume the probability of pigs flying is 10-10

If we find a farm with 100 flying pigs, we conclude our assumption probably wasn’t correct

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Components of aHypothesis Test

Null Hypothesis: H0

Alternative Hypothesis: H1

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Null Hypothesis: H0

The null hypothesis (denoted H0) is a statement that the value of a population parameter (p, µ, σ) isequal to some claimed value.

We test the null hypothesis directly. It will either reject H0 or fail to reject H0

(i.e. accept H0)

Example H0: p = 0.6

H1: p < 0.6

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The alternative hypothesis (denoted H1)

is a statement that the parameter has a value that somehow differs from the null hypothesis.

The difference will be one of <, >, ≠(less than, greater than, doesn’t equal)

Alternative Hypothesis: H1

Example H0: p = 0.6

H1: p < 0.6

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The null hypothesis must say “equal to”:

H0 : p = 0.5

The alternative hypothesis states the difference:

H1 : p > 0.5

Here, the original claim is the alternative hypothesis

Claim: The XSORT method of gender selection increases the likelihood of birthing a girl.

•Let p denote the proportion of girls born.

•The claim is equivilent to “p>0.5”

Example 1

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If we reject the null hypothesis, then the original clam is accepted.

Conclusion: The XSORT method increases the likelihood of having a baby girl.

If we fail to reject the null hypothesis, then the original clam is rejected.

Conclusion: The XSORT method does not increase

the likelihood of having a baby girl.

Claim: The XSORT method of gender selection increases the likelihood of birthing a girl.

Continued

Note: We always test the null hypothesis

Example 1

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Claim: For couples using the XSORT method, the likelihood of having a girl is 50%

•Again, let p denote the proportion of girls born.

•The claim is equivalent to “p=0.5”

The null hypothesis must say “equal to”:

H0 : p = 0.5

The alternative hypothesis states the difference:

H1 : p ≠ 0.5

Here, the original claim is the null hypothesis

Example 2

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If we reject the null hypothesis, then the original clam is rejected.

Conclusion: For couples using the XSORT method, the

likelihood of having a girl is not 0.5

If we fail to reject the null hypothesis, then the original clam is accepted.

Conclusion: For couples using the XSORT method, the

likelihood of having a girl is indeed 0.5

Claim: For couples using the XSORT method, the likelihood of having a girl is 50%

Continued

Note: We always test the null hypothesis

Example 2

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Claim: For couples using the XSORT method, the likelihood of having a girl is at least 50%

•Again, let p denote the proportion of girls born.

•The claim is equivalent to “p ≥ 0.5”

The null hypothesis must say “equal to”:

H0 : p = 0.5

The alternative hypothesis states the difference:

H1 : p < 0.5

Here, the original claim is the null hypothesis

we can’t use ≥ or ≤ in the alternative hypothesis, so we test the negation

Example 3

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Claim: For couples using the XSORT method, the likelihood of having a girl is at least 50%

Continued

If we reject the null hypothesis, then the original clam is rejected.

Conclusion: For couples using the XSORT method, the

likelihood of having a girl is less than 0.5

If we fail to reject the null hypothesis, then the original clam is accepted.

Conclusion: For couples using the XSORT method, the

likelihood of having a girl is at least 0.5Note: We always test the null hypothesis

Example 3

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General rules• If the null hypothesis is rejected,

the alternative hypothesis is accepted. H0 rejected → H1 accepted

• If the null hypothesis is accepted, the alternative hypothesis is rejected. H0 accepted → H1 rejected

• Acceptance or rejection of the null hypothesis is called an initial conclusion.

• The final conclusion is always expressed in terms of the original claim. Not in terms of the null hypothesis or alternative hypothesis.

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Type I Error

• A Type I error is the mistake of rejecting the null hypothesis when it is actually true.

• Also called a “True Negative”True: means the actual hypothesis is trueNegative: means the test rejected the hypothesis

• The symbol (alpha) is used to represent the probability of a type I error.

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Type II Error

• A Type II error is the mistake of acceptingthe null hypothesis when it is actually false.

• Also called a “False Positive”False: means the actual hypothesis is falsePositive: means the test failed to reject the

hypothesis

• The symbol (beta) is used to represent the probability of a type II error.

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Type I and Type II Errors

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Claim: A new medication has greater success rate (p) than that of the old (existing) machine (p0)

•p: Proportion of success for the new medication

•p0: Proportion of success for the old medication

•The claim is equivalent to “p > p0”

Null hypothesis: H0 : p = p0

Alternative hypothesis: H1 : p > p0

Here, the original claim is the null hypothesis

Example 4

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Continued

Type I errorH0 is true, but we reject it → We accept the claim

So we adopt the new (inefficient, potentially harmful) medicine.

(This is called a critical error, must be avoided)

Type II errorH1 is true, but we reject it → We reject the claim

So we decline the new medicine and continue with the old one.

(no direct harm…)

Claim: A new medication has greater success rate (p) than that of the old (existing) machine (p0)

H0 : p = p0

H1 : p > p0

Example 4

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Significance Level

• The probability of a type I error (denoted ) is also called the significance level of the test.

• Characterizes the chance the test will fail.(i.e. the chance of a type I error)

• Used to set the “significance” of a hypothesis test.(i.e. how reliable the test is in avoiding type I errors)

• Lower significance → Lower chance of type I error

• Values used most: = 0.1, 0.05, 0.01 (i.e. 10%, 5%, 1%, just like with CIs)

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Consider a parameter (p, µ, σ, etc.)

The “guess” for the parameter will have a probability that follows a certain distribution (z, t, χ2,etc.)

Note: This is just like what we used to calculate CIs.

Using the significance level α, we determine the region where the guessed value becomes unusual. This is known as the critical region.

The region is described using critical value(s).(Like those used for finding confidence intervals)

Critical Region

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p follows a z-distribution

If we guess p > p0 the critical region is defined by the right tail whose area is α

If we guess p < p0 the critical region is defined by the left tail whose area is α

If we guess p ≠ p0 the critical region is defined by the two tails whose areas are α/2

tα

-tα

-tα/2 tα/2

Example

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1. State the H0 and H1

2. Compute the test statisticDepends on the value being tested

3. Compute the critical region for the test statisticDepends on the distribution of the test statistic (z, t, χ2)

Depends on the significance level α

Found using the critical values

4. Make an initial conclusion from the testReject H0 (accept H1) if the test statistic is within the critical region

Accept H0 if test statistic is not within the critical region

5. Make a final conclusion about the claimState it in terms of the original claim

Testing a Claim Using a Hypothesis Test

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H0 : p = 0.5

H1 : p > 0.5

Claim: The XSORT method of gender selection increases the likelihood of birthing a girl.

Suppose 14 couples using XSORT had 13 girls and 1 boy.

Test the claim at a 5% significance level

1. State H0 and H1

We accept the claim

2. Find the test statistic

3. Find the critical region

4. Initial conclusion

5. Final conclusion

Example 5