copyright 2007 pearson education, inc. publishing as pearson addison-wesley be able to identify...

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Be able to identify the parameter of interest and write both Hypotheses for your test

Be able to check the appropriate conditions for a one-proportion z-test

AP Statistics Objectives Ch21


Be able to calculate the test statistic and P-value for your test

Know when to reject or fail to reject your null hypothesis

Be able to state the conclusion of your test in context



Be able to interpret your P-value

Understand how to use PHANTOMS to preform a one-proportion z-test



Null Hypothesis Alternative Hypothesis Two-sided alternative One-sided alternative P-value One-proportion z-test

Vocabulary


Classroom Notes

Chapter 21 Assignments

Chapter 21 Answers

False Positive – Medicine Tesing

Diagram 2

Diagram 1

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 20- 6

Chapter 20 AssignmentsPart I

Pages 491-492 #6, 8,10,12

Part IIPages 492-493 #14,16, 20, 24


Zero In on the Null 1. How do we choose the null hypothesis?

There is a temptation to state your claim as the null hypothesis. DON’T!

You cannot prove a null hypothesis true.So, it makes more sense to use what you want to show as the alternative.

This way, when you reject the null, you are left with what you want to show.


2. How to Think About P-ValuesA P-value is a conditional probability—the probability of the observed statistic given that the null hypothesis is true.

The P-value is NOT the probability that the null hypothesis is true.

It’s not even the conditional probability that null hypothesis is true given the data.

Be careful to interpret the P-value correctly.


Alpha Levels

3. When the P-value is small, it tells us that our data are rare given the null hypothesis. How rare is “rare”?

We can define “rare event” by setting a arbitrary threshold for our P-value.

If our P-value falls below that point, we’ll reject H0. We call such results statistically significant.The threshold is called an alpha level, denoted by .


Alpha Levels (cont.)4. Common alpha levels are 0.10, 0.05, and 0.01.

The alpha level is also called the significance level.

When we reject the null hypothesis, we say that the test is “significant at that level.”


Alpha Levels (cont.)5. What can you say if the P-value does not fall below ?

You should say that “The data have failed to provide sufficient evidence to reject the null hypothesis.”

Don’t say that you “accept the null hypothesis.”

Recall that, in a jury trial, if we do not find the defendant guilty, we say the defendant is “not guilty”—we don’t say that the defendant is “innocent.”


Alpha Levels (cont.)6. The P-value gives the reader far more information than just stating that you reject or fail to reject the null. In fact, by providing a P-value to the reader, you allow

that person to make his or her own decisions about the test. What you consider to be statistically significant

might not be the same as what someone else considers statistically significant.

There is more than one alpha level that can be used, but each test will give only one P-value.


What Not to Say About Significance7. What do we mean when we say that a test is statistically significant?

All we mean is that the test statistic had a P-value lower than our alpha level.

For large samples, even small, unimportant (“insignificant”) deviations from the null hypothesis can be statistically significant.If the sample is not large enough, even large, financially or scientifically “significant” differences may not be statistically significant.


What Not to Say About Significance (cont.)

8. It’s good practice to report the difference between the observed statistic count and the count which would match the null hypothesis proportion

For example: : = .5 : > .5

With a sample of 45 people out of 80, you would comment on the = .563 being only 5 people more than the 40 out of 80 for a = 0.5


Critical Values Again

9. When making a confidence interval, we’ve found a critical value to correspond to our selected confidence level.

10. Prior to the use of technology, P-values were difficult to find, and it was easier to select a few common alpha values and learn the corresponding critical values for the Normal model.


Critical Values Again (cont.)

11. Rather than looking up your z-score in the table, you could just check it directly against these critical values.

Any z-score larger in magnitude than a particular critical value leads us to reject H0.

Any z-score smaller in magnitude than a particular critical value leads us to fail to reject H0.


Critical Values Again (cont.)12. Here are the traditional critical values from the Normal model:

1-sided 2-sided

0.05 1.645 1.96

0.01 2.28 2.575

0.001 3.09 3.29


Critical Values Again (cont.)

13. When the alternative is one-sided, the critical value puts all of on one side:

14. When the alternative is two-sided, the critical value splits equally into two tails:


Confidence Intervals and Hypothesis Tests15. Confidence intervals and hypothesis tests are built from the same calculations.

They have the same assumptions and conditions.

You can approximate a hypothesis test by examining a confidence interval. Just ask whether the null hypothesis value is

consistent with a confidence interval for the parameter at the corresponding confidence level.


16. Because confidence intervals are two-sided, they correspond to two-sided tests.

In general, a confidence interval with a confidence level of C% corresponds to a two-sided hypothesis test with an -level of 100 – C%.

Slide 21- 20

Confidence Intervals and Hypothesis Tests

𝜶=𝟏𝟎𝟎−𝟗𝟓=𝟓% 95%

_____% CI and a Two-sided Test95%

0.025 0.025


Confidence Intervals and Hypothesis Tests The relationship between confidence intervals an one-sided hypothesis tests is a little more complicated.

17. A confidence interval with a confidence level of C% corresponds to a one-sided hypothesis test with an -level of ½(100 – C)%.

𝜶=𝟏𝟐 (𝟏𝟎𝟎−𝟗𝟎)=𝟓%

90%

_____% CI and a One-sided Test90%

0.05


Making Errors18. Here’s some shocking news for you: nobody’s perfect. Even with lots of evidence we can still make the wrong decision.

19. When we perform a hypothesis test, we can make mistakes in two ways:

I. The null hypothesis is true, but we mistakenly reject it. (Type I error)

II. The null hypothesis is false, but we fail to reject it. (Type II error)


Making Errors 20. Which type of error is more serious depends on the situation at hand. In other words, the gravity of the error is context dependent.


Making Errors

𝜶

21. Here’s an illustration of the four situations in a hypothesis test:

𝜷

Type I Error

Type II Error

Fail to Reject

P(Type I) =

P(Type II) =


22. Medical Testing

Slide 21- 25

What is a False Positive?

What is a False Negative?

Test shows they are not healthy, but they are healthy.

TYPE I Error

Test shows they are healthy, but they are not healthy.

TYPE II Error


1. Public health officials believe that 90% of children have been vaccinated against measles. A random survey of medical records at many schools across the country found that among more than 13,000 children only 89.4% had been vaccinated. A statistician would reject the 90% null hypothesis with a P-value of P = 0.011.

Slide 21- 26

Measles


1. Public health officials believe that 90% of children have been vaccinated against measles. A random survey of medical records at many schools across the country found that among more than 13,000 children only 89.4% had been vaccinated. A statistician would reject the 90% null hypothesis with a P-value of P = 0.011.a) Explain what the P-value means in this context.

Slide 21- 27

There is only a 1.1% chance of seeing a sample proportion of 89.4% or lower vaccinated from sample error, if 90% have really been vaccinated.


1. Public health officials believe that 90% of children have been vaccinated against measles. A random survey of medical records at many schools across the country found that among more than 13,000 children only 89.4% had been vaccinated. A statistician would reject the 90% null hypothesis with a P-value of P = 0.011.a) Explain what the P-value means in this context.b) The result is statistically significant, but is it important? Comment.89.4% indicates 11,622 were vaccinated. While 90% would be 11,700 that were vaccinated. This is f 78

Slide 21- 28

The 90% in the null hypothesis would be 11,700 children and the 89.4% from the sample is 11,622 children. The difference of 0.6% between the two is only 78 children. This may not be important. The large sample size has probably exaggerated the significance.


How often will a Type I error occur? 2. Since a Type I error is rejecting a true null hypothesis,

The probability of a Type I error is our level.

P(Type I Error) =


Making Errors When H0 is false and we reject it, we have done the

right thing.

3. A test’s ability to detect a false hypothesis is called the power of the test. (more to come)


Making Errors4. When H0 is false and we fail to reject it, we have made a Type II error.

is the probability of a Type II error. It’s harder to assess the value of because we

don’t know what the value of the parameter really is.

5. If we decrease the chance of a Type I error, , we increase the chance of a Type II error, .

6. If we increase the chance of a Type I error, , we decrease the chance of a Type II error, .


Making Errors

7. The only way to reduce both types of errors is to collect more data.

8. Otherwise, we just wind up trading off one kind of error against the other.

9. A sample distribution can be estimated using a Normal model when the Independence and Large

Enough Assumptions are satisfied.

p

What happens to this sample distribution when the sample size is increased?

p

10. When the sample size increases, the standard deviation decreases.

p

SD() = SD() =

What happens to this sample distribution when the sample size is decreased?

p

p

11. When the sample size decreases, the standard deviation increases.

SD() = SD() =

12. How does sample size affect Type I Errors?

p

Reject Fail to Reject

is TRUEAs the sample size increases the probability of a Type I Error decreases.

13. How does sample size affect the Type II Errors

p

Reject Fail to Reject

is FALSEAs the sample size increases the probability of a Type II Error decreases.


Power14. POWER – The probability that a hypothesis test will correctly reject a false null hypothesis. Power = 1 -

15. Effect Size – The difference between the null hypothesis, p0, and the true value of a model parameter, p.

16. The larger the EFFECT SIZE, the easier it is to detect a false null hypothesis…the more POWERFUL the test.


A Picture Worth a Thousand Words

17. This diagram shows the relationship between these concepts:

Discuss and the effect size


Reducing Both Type I and Type II Error

Original comparison of errors: Comparison of errors with a larger sample size:


What Can Go Wrong?Don’t interpret the P-value as the probability that H0 is true.

20. The P-value is the probability of the data given that H0 is true, not the other way around.

21. Don’t believe too strongly in arbitrary alpha levels.

22. It’s better to report your P-value and a confidence interval so that the reader can make her/his own decision.

23. Report magnitude of difference too.


What Can Go Wrong?Don’t confuse practical and statistical significance.24. Just because a test is statistically significant doesn’t mean that it is significant in practice.

And, sample size can impact your decision about a null hypothesis, making you miss an important difference or find an “insignificant” difference.

25. Don’t forget that in spite of all your care, you might make a wrong decision.


Chapter 21 Assignment Answers

Slide 21- 47

Part I Part IIPage 491-492 #6, 8 Pages 491 - 492 #10,12

Part III Part IVPage 492 #14,16 Page 493 #20, 24

6 8 10 12

14 16 20 24


Chapter 21 Part I #6

Slide 21- 48

6. A new reading program may reduce the number of elementary students who read below grade level. The company that developed this program supplied materials and teacher training for a large-scale test involving nearly 8500 children in several different school districts. Statistical analysis of the results showed that the percentage of students who did not attain the grade level standard was reduced from 15.9% to 15.1%. The hypothesis that the new reading program produced no improvement was rejected with a P-value of 0.023.



Slide 21- 49


a) Explain what the P-value means in this context.



Slide 21- 50


b) Even though this reading method has been shown to be significantly better, why might you not recommend that your local school adopt it?



Slide 21- 51


b) Even though this reading method has been shown to be significantly better, why might you not recommend that your local school adopt it?



Slide 21- 52

8. Soon after the Euro was introduced as currency in Europe, it was widely reported that someone had spun a Euro coin 250 times and gotten heads 140 times. We wish to test a hypothesis about the fairness of spinning the coin.



Slide 21- 53


a) Estimate the true proportion of heads. Use a 95% confidence interval. Don’t forget to check the conditions first.

The parameter of interest is true proportion of heads when a Euro coin is spun.Independence is reasonable, because coin spins are independent of each other. One spin doesn’t influence the other spins.There were 140 heads and 110 tails. Since both are greater than 10, the sample size is reasonably large enough.



Slide 21- 54


a) Estimate the true proportion of heads. Use a 95% confidence interval. Don’t forget to check the conditions first. Since the conditions are met, we can preform a

one-proportion z-interval.

�̂�=𝟏𝟒𝟎𝟐𝟓𝟎=𝟎 .𝟓𝟔𝑺𝑬 ( �̂� )=√ ( .𝟓𝟔 ) ( .𝟒𝟒 )

𝟐𝟓𝟎 ≈𝟎 .𝟎𝟑𝟏

0.061

0.56 0.061 (0.499 , 0.621)



Slide 21- 55


a) Estimate the true proportion of heads. Use a 95% confidence interval. Don’t forget to check the conditions first.

We are 95% confident that the true proportion of heads when a Euro is spun is between 0.499 and 0.621.



Slide 21- 56


b) Does your confidence interval provide evidence that the coin is unfair when spun? Explain.



Slide 21- 57


c) What is the significance level of this test? Explain.



Slide 21- 58


c) What is the significance level of this test? Explain.



Slide 21- 59

10. The Spike network commissioned a telephone poll of randomly sampled U.S. men. Of the 712 respondents who had children, 22% said “yes” to the question “Are you a stay-at-home dad?” [Time, August 23, 2004]



Slide 21- 60


a) To help them market commercial time, Spike wants an accurate estimate of the true percentage of stay-at-home dads. Construct a 95% confidence interval.The parameter of interest is true proportion of dads stay-at-home dads.



Slide 21- 61


a) To help them market commercial time, Spike wants an accurate estimate of the true percentage of stay-at-home dads. Construct a 95% confidence interval.

Independence is reasonable for the respondents, because

The dads were sampled randomly and712 men is less than 10% of all dads.

They found 157 dads that stay-at-home and 555 that did not. Since both are at least 10, the sample is reasonably large enough.



Slide 21- 62


a) To help them market commercial time, Spike wants an accurate estimate of the true percentage of stay-at-home dads. Construct a 95% confidence interval.Since the conditions are met, we can preform a one-proportion z-interval.



Slide 21- 63



�̂�=𝟎 .𝟐𝟐𝑺𝑬 (�̂� )=√ ( .𝟐𝟐 ) ( .𝟕𝟖 )𝟕𝟏𝟐 ≈𝟎 .𝟎𝟏𝟔

0.031

0.22 0.031 (0.189 , 0.251)



Slide 21- 64



We are 95% confident that the proportion of dads that stay-at-home is between 18.9% and 25.1%.

(0.189 , 0.251)



Slide 21- 65


b) An advertiser of baby-carrying slings for dads will buy commercial time if at least 25% of men are stay-at-home dads. Use your confidence interval to test an appropriate hypothesis and make a recommendation to the advertiser.



Slide 21- 66


c) Could Spike claim to the advertiser that it is possible that 25% of men with young children are stay-at-home dads? What is wrong with the reasoning?



Slide 21- 67


c) Could Spike claim to the advertiser that it is possible that 25% of men with young children are stay-at-home dads? What is wrong with the reasoning?



Slide 21- 68

12. Testing for Alzheimer’s disease can be a long and expensive process, consisting of lengthy tests and medical diagnosis. Recently, a group of researchers (Solomon et al., 1998) devised a 7-minute test to serve as a quick screen for the disease for use in the general population of senior citizens. A patient who tested positive would then go through the more expensive battery of tests and medical diagnosis. The authors reported a false positive rate of 4% and a false negative rate of 8%.



Slide 21- 69


a) Put this in the context of a hypothesis test. What are the null and alternative hypotheses?



Slide 21- 70


b) What would a Type I error mean?



Slide 21- 71


c) What would a Type II error mean?



Slide 21- 72


d) Which is worse here, a Type I or Type II error? Explain.



Slide 21- 73


e) What is the power of this test?

Remember:Power = 1 -



Slide 21- 74


e) What is the power of this test?

Remember:Power = 1 -


Chapter 21 Part II #14

Slide 21- 75

14. Production managers on an assembly line must monitor the output to be sure that the level of defective products remains small. They periodically inspect a random sample of the items produced. If they find a significant increase in the proportion of items that must be rejected, they will halt the assembly process until the problem can be identified and repaired.



Slide 21- 76


a) In this context, what is a Type I error?



Slide 21- 77


b) In this context what is a Type II error?



Slide 21- 78


c) Which type of error would the factory owner consider more serious? Why?



Slide 21- 79


d) Which type of error might customers consider more serious? Why?



Slide 21- 80


d) Which type of error might customers consider more serious? Why?



Slide 21- 81

16. Consider again the task of the quality inspector in question 14.


a) In this context, what is meant by the power of the test the inspectors conduct?



Slide 21- 82



b) They are currently testing 5 items each hour. Someone has proposed they test 10 each hour instead. What are the advantages and disadvantages of such a change.



Slide 21- 83



c) Their test currently uses a 5% level of significance. What are the advantages and disadvantages of changing to an alpha level of 1%?



Slide 21- 84



d) Suppose that as a day passes one of the machines on the assembly line produces more and more items that are defective. How will this affect the power of the test?



Slide 21- 85



d) Suppose that as a day passes one of the machines on the assembly line produces more and more items that are defective. How will this affect the power of the test?



Slide 21- 86

20. A company is willing to renew its advertising contract with a local radio station only if the station can prove that more than 20% of the residents of the city have heard the ad and recognize the company’s product. The radio station conducts a random phone survey of 400 people.



Slide 21- 87


a) What are the hypotheses?



Slide 21- 88


b) The station plans to conduct this test using a 10% level of significance, but the company wants the significance level lowered to 5%. Why?



Slide 21- 89


c) What is meant by the power of this test?



Slide 21- 90


d) For which level of significance will the power of this test be higher? Why?



Slide 21- 91


e) They finally agree to use = 0.05, but the company proposes that the station call 600 people instead of the 400 initially proposed. Will that make the risk of Type II error higher or lower? Explain.



Slide 21- 92


e) They finally agree to use = 0.05, but the company proposes that the station call 600 people instead of the 400 initially proposed. Will that make the risk of Type II error higher or lower? Explain.



Slide 21- 93

24. An artist experimenting with clay to create pottery with a special texture has been experiencing difficulty with these special pieces. About 40% break in the kiln during firing. Hoping to solve this problem, she buys some more expensive clay form another supplier. She plans to make and fire 10 pieces and will decide to use the new clay if at most one of them breaks.



Slide 21- 94


a) Suppose the new expensive clay really is no better than her usual clay. What’s the probability that this test convinces her to use it anyway? (Hint: Use a Binomial model.)



Slide 21- 95


b) If she decides to switch to the new clay and it is no better, what kind of error did she commit?



Slide 21- 96


c) If the new clay really could reduce breakage to only 20%, what’s the probability that her test will not detect the improvement?

Use Binom (10,0.20) Let X = the number of broken pieces out of n = 10.P(at least 2 break) = 1 – P(X < 1) 0.6242

if 2 or more pieces break.



Slide 21- 97


d) How can she improve the power of her test? Offer at least two suggestions.



Slide 21- 98


d) How can she improve the power of her test? Offer at least two suggestions.

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