7



7.2 THE LANGUAGE OF STATISTICAL DECISION MAKING

DEFINITIONS:

The population is the entire group of objects or individuals under study, about which information is desired.

A sample is a part of the population that is actually used to get information.

Statistical inference is the process of drawing conclusions about the population based on information from a sample of that population.

DEFINITIONS:

The null hypothesis, denoted by [pic], is the conventional belief--the status quo, or prevailing viewpoint, about a population.

The alternative hypothesis, denoted by [pic], is an alternative to the null hypothesis – the statement that there is an effect, a difference, a change in the population.

Tip: Having trouble determining the alternative hypothesis? Ask yourself “Why is the research being conducted?” The answer is generally the alternative hypothesis, which is why the alternative hypothesis is often referred to as the research hypothesis.

Aspirin Cuts Cancer Risk Problem 1

According to the American Cancer Society, the lifetime risk of developing colon cancer is 1 in 16. A study suggests that taking an aspirin every other day for 20 years can cut your risk of colon cancer nearly in half. However, the benefits may not kick in until at least a decade of use.

a) Write the null and the alternative hypotheses for this setting.

b) Is this a one-sided to the right, one-sided to the left, or two-sided alternative hypothesis? Hint: look at the alternative hypothesis.

Solution 1

a) [pic]: Taking an aspirin every other day for 20 years will not change the risk of getting colon cancer, which is 1 in 16.

[pic]: Taking an aspirin every other day for 20 years will reduce the risk of getting colon cancer, so the risk will be less than 1 in 16.

b) The alternative hypothesis is one-sided to the left.

Average Life Span Problem 2

Suppose you work for a company that produces cooking pots with an average life span of seven years. To gain a competitive advantage, you suggest using a new material that claims to extend the life span of the pots. You want to test the hypothesis that the average life span of the cooking pots made with this new material increases.

a) Write the null and the alternative hypotheses for this setting.

b) Is this a one-sided to the right, one-sided to the left, or two-sided alternative hypothesis? Hint: look at the alternative hypothesis.

Solution 2

a) [pic]: The average life span of the new cooking pots is 7 years.

[pic]: The average life span of the new cooking pots is greater than 7 years.

b) The alternative hypothesis is one-sided to the right.

Poll Results Problem 3

Based on a previous poll, the percentage of people who said they plan to vote for the Democratic candidate was 50%. The presidential candidates will have daily televised commercials and a final political debate during the week before the election. You want to test the hypothesis that the population proportion of people who say they plan to vote for the Democratic candidate has changed.

a) Write the null and the alternative hypotheses for this setting.

b) Is this a one-sided to the right, one-sided to the left, or two-sided alternative hypothesis? Hint: look at the alternative hypothesis.

Solution 3

a) [pic]: The percentage of the Democratic votes in the upcoming election will be 50%.

[pic]: The percentage of the Democratic votes in the upcoming election will be different from 50%.

b) The alternative hypothesis is two-sided.

Let's Do It! Fair Die?

In a famous die experiment, out of 315,672 rolls, a total of 106,656 resulted in either a ‘5’ or a ‘6’. If the die is "fair”-that is, each of the six outcomes has the same chance of occurring-then the true proportion of 5's or 6's should be 1/3.

However, a close examination of a real die reveals that the “pips” are made by small indentations into the faces of the die. Sides 5 and 6 have more indentations than the other faces, and so these sides should be slightly lighter than the other faces. This suggests that the true proportion of 5’s or 6’s may be a bit higher than the “fair” value, 1/3.

State the appropriate null and alternative hypotheses for assessing if the data provide compelling evidence for the competing theory.

[pic]: The die is fair - that is, the indentations have no effect, and the proportion of 5’s or 6’s is __________________.

Stress can cause sneezes

FROM THE NEW YORK TIMES

Winter can give you a cold because it forces you indoors with coughers, sneezers and wheezers. Toddlers can give you a cold because they are the original Germs “R” Us. But can going postal with the boss or fretting about marriage give a person a post-nasal drip?

Yes, say a growing number of researchers. A psychology professor at Carnegie Mellon University in Pittsburgh, Dr. Sheldon Cohen, said his most recent students suggest that stress doubles a person’s risk of getting a cold.

[pic]: The die is not fair - that is, the indentations do have an effect, and the proportion of 5’s or 6’s is _________________.

Let's Do It! Stress Can Cause Sneezes

Excerpts from the article “Stress can cause sneezes” (The New York Times, January 21, 1997) are shown at the right. Studies suggest that stress doubles a person’s risk of getting a cold. Acute stress, lasting maybe only a few minutes, can lead to colds. One mystery that is still prevalent in cold research is that while many individuals are infected with the cold virus, very few actually get the cold. On average, up to 90% of people exposed to a cold virus become infected, meaning the virus multiplies in the body, but only 40 percent actually become sick. One researcher thinks that the accumulation of stress predisposes an infected person to illness.

The percentage of people exposed to a cold virus who actually get a cold is 40%. The researcher would like to assess if stress increases this percentage. So, the population of interest is people who are under (acute) stress. State the appropriate hypotheses for assessing the researcher’s theory.

[pic]:________________________________________________

Tip: Having trouble determining the alternative hypothesis? Ask yourself "Why is the research being conducted?" The answer is generally the alternative hypothesis, which is why the alternative hypothesis is often referred to as the research hypothesis.

[pic]:_________________________________________________

Example Is the New Drug Better?

Suppose that you have developed a new and very expensive drug intended to cure some disease. You wish to assess how well your new drug performs compared to the standard drug by testing the following hypotheses:

[pic]: The new drug is as effective as the standard drug.

[pic]: The new drug is more effective than the standard drug.

A study is conducted in which the investigator administers the new drug to some number of patients suffering from the disease and the standard drug to another group of patients suffering from the disease. The proportion of cures for both drugs is recorded. Based on this information, we have to decide which hypothesis to support.

a) If the proportion of subjects cured with the new drug was exactly equal to the proportion of subjects cured with the standard drug, which hypothesis would you support?

b) If 75% of the subjects were cured with the new drug while 55% of the subjects were cured with the standard drug, for a difference in cure rates of 20%, which hypothesis would you support? (Define the difference in cure rates as the percent of subjects cured with the new drug less the percent cured with the standard drug.)

c) If the difference in cure rates was equal to 2%, which hypothesis would you support?

(d) How large of a difference in the cure rates is needed for you to feel confident in rejecting the null hypothesis?

What we’ve learned: In real situations the decision maker makes a choice among various alternative courses of actions. According to the consequences of the decision, one researcher might decide to reject the null hypothesis and another one might decide not to reject it.

Let’s Do It! Null and Alternative Hypotheses

State the null and alternative hypothesis that would be used to test the following statements -- these statements are the researcher’s claim, to be stated as the alternative hypothesis. All hypotheses should be expressed in terms of the population mean of interest.

(a) The mean age of patients at a hospital is more than 60 years. [pic] versus [pic]

(b) The mean caffeine content in a cup of regular coffee is less than 110 mg. [pic] versus [pic]

(c) The average number of emergency room admissions per day differs from 20.

[pic] versus [pic]

What Errors Could We Make?

DEFINITION:

Rejecting the null hypothesis [pic] when in fact it is true, is called Type I error.

Failing to reject the null hypothesis [pic] when in fact it is false, is called a Type II error.

[pic]

Remember that ...

A Type I error can only be made if the null hypothesis is true.

A Type II error can only be made if the alternative hypothesis is true.

Let’s Do It! Rain, Rain, Go Away!

You plan to walk to a party this evening. Are you going to carry an umbrella with you? You don’t want to get wet if it should rain. So you wish to test the following hypotheses:

H0: Tonight it is going to rain.

H1: Tonight it is not going to rain.

a) Describe the two types of error that you could make when deciding between these two hypotheses.

b) What are the consequences of making each type of error?

DEFINITION:

The data collected are said to be statistically significant if they are very unlikely to be observed under the assumption that [pic] is true. If the data are statistically significant, then our decision would be to reject [pic].

DEFINITION:

The direction of extreme corresponds to the position of the values that are more likely under the alternative hypothesis [pic] than under the null hypothesis [pic]. If the larger values are more likely under [pic] than under [pic], then the direction of extreme is said to be to the right.

DEFINITION:

The value under the null hypothesis [pic] which is least likely, but at the same time is very likely under the alternative hypothesis [pic] is called the most extreme value.

DEFINITION:

A critical region is the set of values for which you would reject the null hypothesis [pic]. Such values are contradictory to the null hypothesis and favor the alternative hypothesis [pic].

A non-critical region is the set of values for which you would Not Reject the null hypothesis [pic].

The cut-off value or critical value is the value which marks the starting point of the set of values that comprise the rejection region.

Testing Hypothesis about the Mean

The Central Limit Theorem:

The Distribution of [pic] the Sample Mean:

If a simple random sample of size n is taken from a population having population mean [pic] and population standard deviation [pic], or If the sample size n is large enough ([pic]), then all possible sample averages will be symmetric distribution as follows:

[pic]

This symmetric distribution is called the student-t-distribution.

The Student’s t-Distribution with (n-1) degrees of freedom

Properties of the Student’s t-distribution

▪ The t-distribution has a symmetric bell-shaped density centered at 0.

▪ The t-distribution is “flatter” and has “heavier tails” than the N(0,1) distribution.

▪ As the sample size increases, the t-distribution approaches the N(0,1) distribution.

Refer to Table F in you textbook for the t-distribution.

A one-tailed test indicates that the null hypothesis should be rejected when the test value is in the critical region on one side of the mean. A one-tailed test is either a right-tailed test or left-tailed test, depending on the direction of the inequality of the alternative hypothesis.

Example

Finding the Critical Value for [pic]=0.01 (Right-Tailed Test) using The T-distribution with d.f. =120 .

Solution

The critical value is for [pic]=0.01 is 2.358.

Let’s Do It! 5 Critical Values.

a. Finding the Critical Value for [pic]=0.05 (Left-Tailed Test) using the

T-distribution with d.f.=22

b. Finding the Critical Value for [pic]=0.05 (Two-Tailed Test) using the

T-distribution with d.f.=27

Example

A researcher thinks that the average salary of assistant professors is more than $42,000. A sample of 30 assistant professors has a mean salary of $43,260. At [pic]=0.05, test the claim that assistant professors earn more than $42,000 a year. The standard deviation of the sample is $5230.

Step1: State the hypotheses and identify the claim.

[pic] (or [pic])

[pic] ( This is the researcher’s claim

Step2: Find the critical values (the cut off value for an alpha of 0.05).

Since [pic] and the test is a right-tailed test, the critical value is t29, 0.95 =+1.699(This means that any value on the right of the mean that is more than 1.699 standard deviations will be considered extreme under the null hypothesis)

Step3: Compute the standardized t-test statistic and the p-value

[pic] (so, our average of 43,260 is 1.32 standard deviations from 42,000( sample average 43,260is likely to happen under the null H0)

p-value=( Area to the right of 1.32)

= tcdf(1.32, 10^99, 29)

= 0.0986

Step4: Make the decision.

Since the test statistic, +1.32, is less than the critical value, +1.699, and is not in the critical region, the decision is not to reject to reject the null.

Note the p-value 0.0986 > 0.05-( we don’t reject H0.

Step5: Summarize the results

There is not enough evidence to support the claim that assistant professors earn more on average than $42,000 a year.

Let’s Do it! Waste Data

Waste data for a supplier plant for 19 weeks are provided below to assess if, on average, this plant performs worse than the computer at controlling waste. If we let [pic] be the mean percentage of waste for this plant, then we wish to know if the mean[pic] is greater than 0%. The sample standard deviation of 4.40 is reported.

a) State the appropriate hypotheses

b) From the alternative hypothesis, what is the direction of extreme in this case?

c) Standardize the observed sample mean under Ho and to obtain the observed t-test statistic. And Compute the p-value.

d) Using a significance level (α=0.01), what would be your decision?

e) Below is the SPSS output for a one-sample t-test. Verify you work above.

[pic]

From the output we have an observed T-test statistic of t = 4.783 and a two-sided p-value of 0.00015.Our one-sided p-value to the right of 4.783 will be half of the

2-sided value, p-value = 0.00015/2 = 0.000075.

Let's Do It!.4 pH Levels

A soil scientist is interested in studying the pH level in the soil for a certain field. In particular, she is going to examine a random sample of soil samples and measure their pH levels to assess if the mean field pH level is neutral (that is, equal to 7) versus the alternative hypothesis that the mean pH level is acidic (that is, less than 7). She observed the following pH levels in the sample: 5.8, 6.3, 6.9, 6.2, 5.5.

(a) State the corresponding null and alternative hypotheses to be tested.

(b) Find sample mean pH level and sample standard deviation.

(c) Compute the observed standardized test statistic and find the corresponding p-value.

(d) At a level of significance [pic] 5%. Is the result statistically significant? Explain.

(e) State your conclusion using a well-written sentence.

Let's Do It! Market Value for Homes

A real-estate appraiser wants to verify the market value for homes on the east side of the city that are very similar in size and style. The appraiser wants to test the popular belief that the average sales price is $37.80 per square foot for such homes versus that the average differs from $37.80. He wants to use a significance level of 0.01.

(a) State the appropriate null and alternative hypotheses about μ, the population mean sales price for such homes.

(b) Suppose that the random sample of six sales were selected. The sampled sales prices per square foot are $35.00, $38.10, $30.30, $37.20, $29.80, and $35.40. Does it appear that the popular perception of the market value is valid for this neighborhood? Give the value of the standardized t- test value and the p-value.

(c) Based on your p-value in (b), are the results statistically significant at a 1% significance level? Explain.

Homework page258: 1-4all ,33-35, 51,52, 93,94 .

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p. 5 - 6

p. 6

Definition

The critical value separates the critical region from the noncritical region. The symbol for critical value is C.V.

The critical or rejection region is the range of values of the test value that indicates that there is a significant difference and that the null hypothesis should be rejected.

The noncritical or non-rejection region is the range of values of the test value that indicates that the difference was probably due to chance and that the null hypothesis should not be rejected.

Hypothesis Testing

In order to perform the hypothesis test, many hypotheses are tested using a statistical test based on the following general formula:

[pic]

In particular, when testing the population mean, our test value is the standardized T-Test value [pic]

Hypothesis-Testing Problems (Traditional Method)

Step1: State the null and the Alternative and identify the claim.

Step2: Find the critical value(s) from the appropriate table.

Step3: Compute the test value and the p-value.

Step 4: Summarize the results. Note: Never accept the null. We reserve the judgment about the hypothesis is true.

Summary of the One-Sample t-Test for a Population Mean:

We were interested in testing hypotheses about the population mean [pic]. The null hypothesis is [pic] where [pic] is the hypothesized value for [pic]. The alternative hypothesis provides the direction for the test. These hypotheses are statements about the population mean, not the sample mean. The significance level að is selected.

▪ We base our decision about [pic] on the standardized sample mean, which is

[pic].

This is the T- test statistic, and the distributα is selected.

We base our decision about [pic] on the standardized sample mean, which is

[pic].

This is the T- test statistic, and the distribution of the variable T under [pic] is a t-distribution with n-1 degrees of freedom.

Note: The test statistic is the same no matter how the alternative hypothesis is expressed.

▪ We calculate the p-value for the test, which depends on how the alternative hypothesis is expressed (see below).

One-sided, to the right One-sided, to the Left Two-sided

If [pic], then the If [pic], then the If [pic], then the

p-value is the area to the right p-value is the area to the left p-value is the area in the two tails,

of the observed test statistic of the observed test statistic outside of the observed test

under the [pic] model. under the [pic] model. statistic under the [pic] model

[pic] [pic] [pic]

▪ Decision: A p-value less than the significance level leads to rejection of [pic].

Note: The TI graphing calculator has a STAT TESTS menu that allows you to perform hypothesis tests for a population mean. See the TI Quick Steps appendix at the end of this chapter for details.

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