Example 10.1 Experimenting with a New Pizza Style at the Pepperoni Pizza Restaurant

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Example 10.1Experimenting with a New Pizza Style at the Pepperoni Pizza Restaurant

Concepts in Hypothesis Testing


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Background Information

• The manager of Pepperoni Pizza Restaurant

has recently begun experimenting with a new

method of baking its pepperoni pizzas.


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Background Information – cont’d

• He believes that the new method produces a

better-tasting pizza, but he would like to base a

decision on whether to switch from the old

method to the new method on customer

reactions.

• Therefore he performs an experiment.


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The Experiment

• For 100 randomly selected customers who

order a pepperoni pizza for home delivery, he

includes both an old style and a free new style

pizza in the order.


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The Experiment – cont’d

• All he asks is that these customers rate the

difference between pizzas on a -10 to +10

scale, where -10 means they strongly favor the

old style, +10 means they strongly favor the

new style, and 0 means they are indifferent

between the two styles.

• Once he gets the ratings from the customers,

how should he proceed?


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Hypothesis Testing

• This example’s goal is to explain hypothesis

testing concepts. We are not implying that

the manager would, or should, use a

hypothesis testing procedure to decide

whether to switch methods.


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Hypothesis Testing – cont’d

• First, hypothesis testing does not take costs

into account. In this example, if the new method

is more costly it would be ignored by hypothesis

testing.

• Second, even if costs of the two pizza-making

methods are equivalent, the manager might

base his decision on a simple point estimate

and possibly a confidence interval.


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Null and Alternative Hypotheses

• Usually, the null hypothesis is labeled Ho and

the alternative hypothesis is labeled Ha.

• The null and alternative hypotheses divide all

possibilities into two nonoverlapping sets,

exactly one of which must be true.


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Null and Alternative Hypotheses – cont’d

• Traditionally, hypotheses testing has been

phrased as a decision-making problem, where

an analyst decides either to accept the null

hypothesis or reject it, based on the sample

evidence.


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One-Tailed Versus Two-Tailed Tests

• The form of the alternative hypothesis can be

either a one-tailed or two-tailed, depending

on what the analyst is trying to prove.

• A one-tailed hypothesis is one where the only

sample results which can lead to rejection of

the null hypothesis are those in a particular

direction, namely, those where the sample

mean rating is positive.


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One-Tailed Versus Two-Tailed Tests – cont’d

• A two-tailed test is one where results in either of

two directions can lead to rejection of the null

hypothesis.

• Once the hypotheses are set up, it is easy to

detect whether the test is one-tailed or two-

tailed.


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One-Tailed Versus Two-Tailed Tests – cont’d

• One tailed alternatives are phrased in terms of “>”

or “<“ whereas two tailed alternatives are phrased

in terms of “”

• The real question is whether to set up hypotheses

for a particular problem as one-tailed or two-

tailed.

• There is no statistical answer to this question. It

depends entirely on what we are trying to prove.


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Types of Errors

• Whether or not one decides to accept or reject

the null hypothesis, it might be the wrong

decision.

• One might reject the null hypothesis when it is

true or incorrectly accept the null hypothesis

when it is false.

• These errors are called type I and type II errors.


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Types of Errors – cont’d

• In general we incorrectly reject a null

hypothesis that is true. We commit a type II

error when we incorrectly accept a null

hypothesis that is false.

• These ideas appear graphically below.


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Types of Errors -- continued

• While these errors seem to be equally

serious, actually type I errors have

traditionally been regarded as the more

serious of the two.

• Therefore, the hypothesis-testing procedure

factors caution in terms of rejecting the null

hypothesis.


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Significance Level and Rejection Region

• The real question is how strong the evidence in

favor of the alternative hypothesis must be to

reject the null hypothesis.

• The analyst determines the probability of a type

I error that he is willing to tolerate. The value is

denoted by and is most commonly equal to

0.05, although sigma=0.01 and sigma=0.10 are

also frequently used.


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Significance Level and Rejection Region – cont’d

• The value of is called the significance level

of the test.

• Then, given the value of sigma, we use

statistical theory to determine the rejection

region.


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Significance Level and Rejection Region – cont’d

• If the sample falls into this region we reject the

null hypothesis; otherwise, we accept it.

• Sample evidence that falls into the rejection

region is called statistically significant at the

sigma level.


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Significance from p-values

• This approach is currently more popular than

the significance level and rejected region

approach.

• This approach is to avoid the use of the level

and instead simply report “how significant” the

sample evidence is.


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Significance from p-values – cont’d

• We do this by means of the p-value.The p-

value is the probability of seeing a random

sample at least as extreme as the sample

observes, given that the null hypothesis is true.

• Here “extreme” is relative to the null hypothesis.


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• In general smaller p-values indicate more

evidence in support of the alternative

hypothesis. If a p-value is sufficiently small,

almost any decision maker will conclude that

rejecting the null hypothesis is the more

“reasonable” decision.


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• How small is a “small” p-value? This is largely a matter of semantics but if the − p-value is less than 0.01, it provides

“convincing” evidence that the alternative hypothesis is true;

− p-value is between 0.01 and 0.05, there is “strong” evidence in favor of the alternative hypothesis;


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− p-value is between 0.05 and 0.10, it is in a “gray area”;

− p-values greater than 0.10 are interpreted as weak or no evidence in support of the alternative.

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Example 10.1 Experimenting with a New Pizza Style at the Pepperoni Pizza Restaurant