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Chapter 10:

Section 10.1:

Vocabulary of Hypothesis Testing

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Testing Effects:

We have developed a new drug, or a new procedure, a new technique, additive, or method; something that we believe will have an effect on a known and understood event.

We need to test and see if there really is an effect or not. One Method for this is called Null Hypothesis testing.

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Steps for Null Hypothesis Testing:

Null Hypothesis Testing is a method of using data to summarize the evidence about a hypothesis

A Null Hypothesis Test has three steps

1. Statement:1. Determine the Null Hypothesis

2. Determine the alternative Hypothesis

2. Evidence:1. Collect Data from the population in order to test the statement.

3. Analysis1. Examine the data to make conclusions about the statement.

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Pre-Step: Assumptions:

We will assume all data is collected randomly without significant bias. If this assumption does not hold NO reasonable conclusions can be made.

We will also assume the sample sizes are large enough to satisfy our past conditions. Again if this assumption does not hold we can not consider our findings meaningful.

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Step one: The Hypotheses:

A hypothesis is a statement about a population, usually a particular value or range of values for a parameter.

Our goal will be to see if the data supports the hypothesis.

Example 1: The standard method works 43% of the time.

Example 2: Psychics are right more then 50% of the time.

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Step One (continued): The Null Hypothesis:

Each test has two hypotheses:

The null hypothesis is a statement that the parameter takes a particular value.

The alternative hypothesis states that the parameter falls in some alternative range of values.

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“Null?” The value in the null hypothesis represents no effect

The symbol Ho denotes null hypothesis

The null hypothesis is assumed true unless proven otherwise.

It will be a statement about a populations parameter. Mean Proportion Standard deviation

ALWAYS A STATEMENT OF EQUALITY If you guess you will be right 50% of the time. This type of treatment yields an average drop of 32 HDL points The Standard deviation of this machinery’s output is 2.45 units.

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“Alternitive” The range of the Alternative Hypothesis represents the

effect we are trying to support. The symbol Ha denotes alternative hypothesis

The alternative hypothesis must be supported by data

It will be a statement about a populations parameter. Mean Proportion Standard deviation

ALWAYS A RANGE OF VALUES AROUND NULL HYPO Psychics are right more then 50% of the time. The new treatment drops HDLs more then 32 points The new equipment decrease the standard deviation of our outputs

below 2.45 units.

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In this chapter, there are three ways to set up the null and alternative hypotheses:

1. Equal versus not equal hypothesis (two-tailed test)

H0: parameter = some value

H1: parameter ≠ some value

2. Equal versus less than (left-tailed test)

H0: parameter = some value

H1: parameter < some value

3. Equal versus greater than (right-tailed test)

H0: parameter = some value

H1: parameter > some value

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For each of the following claims, determine the null and alternative hypotheses. State whether the test is two-tailed, left-tailed or right-tailed.

a) In 2008, 62% of American adults regularly volunteered their time for charity work. A researcher believes that this percentage is different today.

b) According to a study published in March, 2006 the mean length of a phone call on a cellular telephone was 3.25 minutes. A researcher believes that the mean length of a call has increased since then.

c) Using an old manufacturing process, the standard deviation of the amount of wine put in a bottle was 0.23 ounces. With new equipment, the quality control manager believes the standard deviation has decreased.

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Solution

a) In 2008, 62% of American adults regularly volunteered their time for charity work. A researcher believes that this percentage is different today.

The hypothesis deals with a population proportion, p. If the percentage participating in charity work is no different than in 2008, it will be 0.62 so the null hypothesis is H0: p=0.62.

Since the researcher believes that the percentage is different today, the alternative hypothesis is a two-tailed hypothesis: H1: p≠0.62.

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Solution

b) According to a study published in March, 2006 the mean length of a phone call on a cellular telephone was 3.25 minutes. A researcher believes that the mean length of a call has increased since then.

The hypothesis deals with a population mean, . If the mean call length on a cellular phone is no different than in 2006, it will be 3.25 minutes so the null hypothesis is H0: =3.25.

Since the researcher believes that the mean call length has increased, the alternative hypothesis is: H1: > 3.25, a right-tailed test.

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Solution

c) Using an old manufacturing process, the standard deviation of the amount of wine put in a bottle was 0.23 ounces. With new equipment, the quality control manager believes the standard deviation has decreased.

The hypothesis deals with a population standard deviation, . If the standard deviation with the new equipment has not changed, it will be 0.23 ounces so the null hypothesis is H0: = 0.23.

Since the quality control manager believes that the standard deviation has decreased, the alternative hypothesis is: H1: < 0.23, a left-tailed test.

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Questions?

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What can go Wrong??

Type One Errors;

Type Two Errors;

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The Four things that can happen

1. We reject the null hypothesis when the alternative hypothesis is true. This decision would be correct.

2. We do not reject the null hypothesis when the null hypothesis is true. This decision would be correct.

3. We reject the null hypothesis when the null hypothesis is true. This decision would be incorrect. This type of error is called a Type I error.

4. We do not reject the null hypothesis when the alternative hypothesis is true. This decision would be incorrect. This type of error is called a Type II error.

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What can go Wrong??

Type One Errors; We think there was an effect when there was not. We falsely reject the null hypothesis.

Type Two Errors; We think there was no effect when there was. We fail to reject the null hypothesis.

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For each of the following claims, explain what it would mean to make a Type I error. What would it mean to make a Type II error?

a) In 2008, 62% of American adults regularly volunteered their time for charity work. A researcher believes that this percentage is different today.

b) According to a study published in March, 2006 the mean length of a phone call on a cellular telephone was 3.25 minutes. A researcher believes that the mean length of a call has increased since then.

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a) In 2008, 62% of American adults regularly volunteered their time for charity work. A researcher believes that this percentage is different today.

A Type I error is made if the researcher concludes that p≠0.62 when the true proportion of Americans 18 years or older who participated in some form of charity work is currently 62%.

A Type II error is made if the sample evidence leads the researcher to believe that the current percentage of Americans 18 years or older who participated in some form of charity work is still 62% when, in fact, this percentage differs from 62%.

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b) According to a study published in March, 2006 the mean length of a phone call on a cellular telephone was 3.25 minutes. A researcher believes that the mean length of a call has increased since then.

A Type I error occurs if the sample evidence leads the researcher to conclude that >3.25 when, in fact, the actual mean call length on a cellular phone is still 3.25 minutes.

A Type II error occurs if the researcher fails to reject the hypothesis that the mean length of a phone call on a cellular phone is 3.25 minutes when, in fact, it is longer than 3.25 minutes.

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Risk of having Errors;

We will ALWAYS be at risk of both types of Error.

The Probability of making a type one Error is called the Level of Significance for that test.

This Level of Significance is denoted α.

This Significance level is chosen before any data is collected.

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= P(Type I Error)

= P(rejecting H0 when H0 is true) = P(Guilty when Innocent)

= P(Type II Error)

= P(not rejecting H0 when H1 is true) = P(Not Guilty when Guilty)

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Warning!!

As the probability of a Type I error increases, the probability of a Type II error decreases, and vice-versa.

We say “α varies inversely with β”

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Practice Stating conclutions

1) According to a study published in March, 2006 the mean length of a phone call on a cellular telephone was 3.25 minutes. A researcher believes that the mean length of a call has increased since then.

Suppose the sample evidence indicates that the null hypothesis should be rejected. State the wording of the conclusion.

Suppose the sample evidence indicates that the null hypothesis should not be rejected. State the wording of the conclusion.

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Suppose the sample evidence indicates that the null hypothesis should be rejected. State the wording of the conclusion.

The statement in the alternative hypothesis is that the mean call length is greater than 3.25 minutes. Since the null hypothesis (=3.25) is rejected, we conclude that;

There is sufficient evidence to conclude that the mean length of a phone call on a cell phone is greater than 3.25 minutes.

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Suppose the sample evidence indicates that the null hypothesis should not be rejected. State the wording of the conclusion.Since the null hypothesis (=3.25) is not rejected, we conclude that;

There is insufficient evidence to conclude that the mean length of a phone call on a cell phone is greater than 3.25 minutes.

The sample evidence is consistent with the mean call length equaling 3.25 minutes.