Chapter 9: Testing a Claim

Transcription

The Practice of Statistics (4th Edition) - Starnes, Yates, Moore
Chapter 9: Testing a Claim
9.1 Significance Tests: The Basics (pp.528-543)
Read Summary on p. 545 first.
1. What is a significance test?
2. What is the difference between a null and an alternative hypothesis? What notation is used
for each? What is a common mistake when stating hypotheses?
IN CLASS: For each scenario, define the parameter of interest and state appropriate
hypotheses.
(a) Mike is an avid golfer who would like to improve his play. A friend suggests getting new
clubs and lets Mike try out his 7-iron. Based on years of experience, Mike has established that
the mean distance that balls travel when hit with his old 7-iron is  = 175 yards with a standard
deviation of  = 15 yards. He is hoping that this new club will make his shots with a 7-iron
more consistent (less variable), and so he goes to the driving range and hits 50 shots with the
new 7-iron.
(b) At the Hawaii Pineapple Company, managers are interested in the sizes of the pineapples
grown in the company’s fields. Last year, the mean weight of the pineapples harvested from one
large field was 31 ounces. A different irrigation system was installed in this field after the
growing season. Managers wonder if this change will affect the mean weight of pineapples
grown in the field this year.
3. Explain the differences between one-sided and two-sided hypotheses. How can you decide
which one to use?
4. What form does the null and alternative hypothesis take in significance testing?
5. Hypotheses always refer to a ___________, not to a ______________.
6. In statistics, what is meant by the P-value? What does a P-value measure?
IN CLASS: Example: A better golf club?
When Mike was testing a new 7-iron, the hypotheses were H 0 :  = 15 versus H a :  < 15
where  = the true standard deviation of the distances Mike hits golf balls using the new 7-iron.
Based on a sample of shots with the new 7-iron, the standard deviation was s x = 13.9 yards. A
significance test using the sample data produced a P-value of 0.28. Interpret the P-value in this
context.
7. Complete Check Your Understanding on p. 532.
8. If a P-value is small, what do we conclude about the null hypothesis?
9. If a P-value is large, what do we conclude about the null hypothesis?
10. What are common errors students make in their conclusions of P-values?
11. On what evidence would we reject the null hypothesis?
12. On what evidence would we fail to reject the null hypothesis?
13. What is meant by a significance level?
14. Explain what it means to say that data are statistically significant.
15. How small should the P-value be in order to claim that a result is statistically significant?
16. When using a fixed significance level to draw a conclusion in a statistical test what can be
concluded when the P value is  ? ?
IN CLASS: Example Tasty chips
For his second semester project in AP Statistics, Zenon decided to investigate whether students
at his school prefer name-brand potato chips to generic potato chips. After collecting data, Zenon
performed a significance test using the hypotheses H 0 : p = 0.5 versus H a : p > 0.5 where p = the
true proportion of students at his school who prefer name-brand chips. The resulting P-value was
0.074. What conclusion would you make at each of the following significance levels?
(a)  = 0.10
(b)  = 0.05
17. When should a significance level be chosen?
18. Unfortunately, sometimes our conclusions are wrong.
What is a Type I Error?
19. What is a Type II Error ?
20. Which error is worse, Type I or Type II?
21. Complete the Check Your Understanding on page 539.
22. What is the relationship between the significance level  and the probability of Type I Error?
23. How can we reduce the probability of a Type I error?
24. What is meant by the power of a significance test?
25. What is the relationship between Power and Type II Error? Will you be expected to calculate
the power on the AP exam?
26. How can we increase the power of a test?
9.2 Tests about a Population Proportion (pp.549-561)
1. Summarize the three conditions that must be checked before carrying out significance tests:



2. State the general form of the ―test statistic‖.
3. What does the test statistic measure? Is this formula on the AP exam formula sheet?
4. Describe the 4-step process for significance tests. Explain what is required at each step.

State

Plan

Do

Conclude
5. What test statistic is used when testing for a population proportion? Is this on the formula
sheet?
IN CLASS: Complete Check Your Understanding on p. 555.
6. If asked to carry out a signifigance test and there is no  provided, what is recommended?
IN CLASS: Example of Two-Sided Test
7. Can you use confidence intervals to decide between two hypotheses? What is the advantage
of using confidence intervals for this purpose?
8. Why don't we always use confidence intervals?
9.3 Tests about a Population Mean (pp.565-585)
1. What are the three conditions for conducting a significance test for a population mean?
2. What test statistic do we use when testing a population mean? Is this formula on the AP
exam formula sheet?
3. How do you calculate p-values using the t-distributions?
IN CLASS: Example: Construction zones
Every road has one at some point—construction zones that have much lower speed limits. To see
if drivers obey these lower speed limits, a police officer used a radar gun to measure the speed
(in miles per hour, or mph) of a random sample of 10 drivers in a 25 mph construction zone.
Here are the results:
27
33
32
21
30
30
29
25
27
34
(a) Is there convincing evidence that the average speed of drivers in this construction zone is
greater than the posted 25 mph speed limit?
Collection 1
Dot Plot
20 22 24 26 28 30 32 34 36
Speed (miles per hour)
(b) Given your conclusion in part (a), which kind of mistake—a Type I or a Type II error—could
you have made? Explain what this mistake means in this context.
READ pp. 574 – 576
IN CLASS: Two-sided Test for : Example: Don’t Break the Ice
In the children’s game Don’t Break the Ice, small plastic ice cubes are squeezed into a square
frame. Each child takes turns tapping out a cube of ―ice‖ with a plastic hammer, hoping that the
remaining cubes don’t collapse. For the game to work correctly, the cubes must be big enough so
that they hold each other in place in the plastic frame but not so big that they are too difficult to
tap out. The machine that produces the plastic cubes is designed to make cubes that are 29.5
millimeters (mm) wide, but the actual width varies a little. To ensure that the machine is working
well, a supervisor inspects a random sample of 50 cubes every hour and measures their width.
One sample produced a mean width of 29.4943 mm with a standard deviation of 0.0877 mm.
(a) Do these data give convincing evidence that the mean width of cubes produced this hour is
not 29.5 mm? Use a significance test with  = 0.05 to find out.
(b) Calculate a 95% confidence interval for  . Does your interval support your decision from
part (a)?
4. In terms of rejecting the hypothesis H 0 , how is a significance test related to a confidence
interval on the same population?
5. Use your calculator to find the p value (tcdf command) for the example Healthy Streams.
What is that p-value?
6. When using technology for the "DO" part of the four step process, what is recommended
on page 573?
7. Work through the Juicy Pineapple example on page 574. Use a calculator to find the
exact P-value. Why is tcdf multiplied by 2?
8. What is the difference between using the calculator versus Table B when finding the pvalue in this example?
9. Do we have encough evidence to reject H 0 in the Juicy Pineapple example? Explain.
10. Read the Check Your Understanding on page 577 and answer questions 1 and 2.
11. What is paired data?
12. What information would lead us to apply a paired t-test to a study, and what would be the
statistic of interest?
13. In the example, Is Caffeine Dependence Real, explain the difference in the "Do"
procedures for this example versus the Juicy Pineapple example.

Chapter 9: Testing a Claim

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