1 chapter9 hypothesis tests using a single sample

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Chapter9Hypothesis Tests

Using a Single Sample

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BASICS

In statistics, a hypothesis is a statement about a population characteristic.

NEVER about a statistic!!!

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FORMAL STRUCTURE

Hypothesis Tests are based on an reductio ad absurdum form of argument.

Specifically, we make an assumption and then attempt to show that assumption leads to an absurdity or contradiction.

Hence, the assumption is wrong.

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FORMAL STRUCTURE

The null hypothesis, denoted H0 is a statement or claim about a population characteristic that is

initially assumed to be true.

The null hypothesis is so named because it is the “starting point” for the investigation. The

phrase “there is no difference” is often used in its interpretation.

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FORMAL STRUCTUREThe alternate hypothesis denoted

by Ha ) is the competing claim. (I call it the RESEARCH hypothesis.)

The alternate hypothesis is a statement about the same population characteristic that is used

in the null hypothesis.

Generally, the alternate hypothesis is a statement that specifies that the population has a value different, in some way, from the

value given in the null hypothesis.

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FORMAL STRUCTURE

Rejection of the null hypothesis will imply the acceptance of this alternative

hypothesis.

Assume H0 is true and attempt to show this leads to an absurdity, therefore H0 is

false and Ha is true.

(Remember proof by contradiction?)

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FORMAL STRUCTURE

Typically one assumes the null hypothesis to be true and then one of the following conclusions are drawn.

1. Reject H0 Equivalent to saying that Ha is correct or true

2. Fail to reject H0 Equivalent to saying that we have failed to show a statistically significant deviation from the claim of the null hypothesisThis is not the same as saying that the null hypothesis is true.

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AN ANALOGY

The Statistical Hypothesis Testing process can be compared very closely with a judicial trial.

1.Assume a defendant is innocent (H0)

2.Present evidence to show guilt

3.Decision: the defendant cannot be innocent

given the evidence

4. Reject the assumption of innocence in

favor of the alternate—guilty (Ha)

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AN ANALOGY

Two Hypotheses are then created.

H0: Innocent

Ha: Not Innocent (Guilt)

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Comments on Hypothesis Form

The null hypothesis must contain the equal sign.

This is absolutely necessary because the test requires the null hypothesis to be assumed to be true. The numeric value attached to the equal sign is then the value assumed to be true and used in subsequent calculations.

The alternate hypothesis should be what you are really attempting to show to be true.

This is not always possible.

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Hypothesis Form

The form of the null hypothesis isH0: population characteristic = hypothesized value where the hypothesized value is a specific number determined by the problem context.

The alternative (or alternate) hypothesis will have one of the following three forms:

Ha: population characteristic > hypothesized value

Ha: population characteristic < hypothesized value

Ha: population characteristic hypothesized value

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Examples of Hypotheses

You would like to determine if the

diameters of the ball bearings you produce have a mean of 6.5 cm.

H0: µ=6.5

Ha: µ≠6.5

(Two-sided alternative)

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The students entering into the math program used to have a

mean SAT quantitative score of 525. Are the current students weaker (as measured by the

SAT)?

H0: µ = 525

Ha: µ < 525(One-sided alternative)


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Do the “16 ounce” cans of peaches canned and sold by DelMonte meet the claim on the label (on the average)?

H0: µ = 16 oz

Ha: µ< 16 oz


Notice, the real concern would be selling the consumer less than 16 ounces of peaches. We don’t really care if the cans are over 16 ozs!

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Is the proportion of defective parts produced by a

manufacturing process more than 5%?

H0: p = 0.05

Ha: p > 0.05


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Do two brands of light bulb have the

same mean lifetime?

H0: µBrand A = µBrand B

Ha: µBrand A µBrand B


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Do parts produced by two different milling machines have the same variability in

diameters?

or equivalently

0 1 2

a 1 2

H : H :

2 20 1 2

2 2a 1 2

H : H :


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Caution

When you set up a hypothesis test, the result is either

1. Strong support for the alternate hypothesis (if the null hypothesis is rejected)

2. There is not sufficient evidence to refute the claim of the null hypothesis. (You are stuck with it, because there is a lack of strong evidence against the null hypothesis.)

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P-value

The P-value is the probability of obtaining a test statistic value as extreme (or more so) assuming that H0 is true!

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The lower the probability of your sample results happening by

chance when Ho is true then the less likely Ho actually is true!

1 chapter9 hypothesis tests using a single sample

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