1. Pull 2 samples of 20 pennies and record both averages (2 dots).
We are going to discuss 9.1 again today because this is a very important section!

2. Chapter 9
Testing a Claim
Section 9.1
Significance Tests: The Basics

3. Significance Tests: The Basics
STATE appropriate hypotheses for a significance test about a population parameter.
INTERPRET a P-value in context.
MAKE an appropriate conclusion for a significance test.
INTERPRET a Type I error and a Type II error in context. GIVE a consequence of each error in a given setting.

4. Stating Hypotheses
Confidence intervals are one of the two most common methods of statistical inference.

5. Stating Hypotheses
Confidence intervals are one of the two most common methods of statistical inference.
The second common method of inference, called a significance test, allows us to weigh the evidence in favor of or against a particular claim.

6. Stating Hypotheses
Confidence intervals are one of the two most common methods of statistical inference.
The second common method of inference, called a significance test, allows us to weigh the evidence in favor of or against a particular claim.
A significance test (or a hypothesis test) is a formal procedure for using observed data to decide between two competing claims (called hypotheses). The claims are usually statements about a parameter, like the population proportion p or the population mean µ.

7. Stating Hypotheses
The claim that we weigh evidence against in a significance test is called the null hypothesis (H0).
The claim that we are trying to find evidence for is the alternative hypothesis (Ha).

8. Stating Hypotheses
The claim that we weigh evidence against in a significance test is called the null hypothesis (H0).
The claim that we are trying to find evidence for is the alternative hypothesis (Ha).
A free-throw shooter claims that his long-run proportion of made free throws is p = 0.8, but we suspect that the player is exaggerating.

9. Stating Hypotheses H0: p = 0.80 Ha: p < 0.80
The claim that we weigh evidence against in a significance test is called the null hypothesis (H0).
The claim that we are trying to find evidence for is the alternative hypothesis (Ha).
A free-throw shooter claims that his long-run proportion of made free throws is p = 0.8, but we suspect that the player is exaggerating.
H0: p = 0.80
Ha: p < 0.80
Usually, the null hypothesis H0 is a statement of “no difference.”

10. Stating Hypotheses
The alternative hypothesis is one-sided if it states that a parameter is greater than the null value or if it states that the parameter is less than the null value.
The alternative hypothesis is two-sided if it states that the parameter is different from the null value (it could be either greater than or less than).

11. Stating Hypotheses H0: p = 0.80 Ha: p < 0.80 Ha: p > 0.80
The alternative hypothesis is one-sided if it states that a parameter is greater than the null value or if it states that the parameter is less than the null value.
The alternative hypothesis is two-sided if it states that the parameter is different from the null value (it could be either greater than or less than).
H0: p = 0.80
Ha: p < 0.80
Ha: p > 0.80

12. Stating Hypotheses H0: p = 0.80 Ha: p < 0.80 Ha: p > 0.80
The alternative hypothesis is one-sided if it states that a parameter is greater than the null value or if it states that the parameter is less than the null value.
The alternative hypothesis is two-sided if it states that the parameter is different from the null value (it could be either greater than or less than).
H0: p = 0.80
Ha: p < 0.80
Ha: p > 0.80

13. Stating Hypotheses H0: p = 0.80 Ha: p < 0.80 Ha: p > 0.80
The alternative hypothesis is one-sided if it states that a parameter is greater than the null value or if it states that the parameter is less than the null value.
The alternative hypothesis is two-sided if it states that the parameter is different from the null value (it could be either greater than or less than).
H0: p = 0.80
Ha: p < 0.80
Ha: p > 0.80
H0: p = 0.80
Ha: p ≠ 0.80

14. Stating Hypotheses H0: p = 0.80 Ha: p < 0.80 Ha: p > 0.80
The alternative hypothesis is one-sided if it states that a parameter is greater than the null value or if it states that the parameter is less than the null value.
The alternative hypothesis is two-sided if it states that the parameter is different from the null value (it could be either greater than or less than).
H0: p = 0.80
Ha: p < 0.80
Ha: p > 0.80
H0: p = 0.80
Ha: p ≠ 0.80

15. Stating Hypotheses
JTB Photo/JTB Photo/Superstock
Problem: At the Hawaii Pineapple Company, managers are interested in the size 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.
State appropriate hypotheses for performing a significance test. Be sure to define the parameter of interest.

16. Stating Hypotheses
JTB Photo/JTB Photo/Superstock
Problem: At the Hawaii Pineapple Company, managers are interested in the size 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.
State appropriate hypotheses for performing a significance test. Be sure to define the parameter of interest.
H0: µ = 31
Ha: µ ≠ 31

17. Stating Hypotheses
JTB Photo/JTB Photo/Superstock
Problem: At the Hawaii Pineapple Company, managers are interested in the size 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.
State appropriate hypotheses for performing a significance test. Be sure to define the parameter of interest.
H0: µ = 31
Ha: µ ≠ 31
where µ = the true mean weight of all pineapples grown in the field this year.

18. Stating Hypotheses
CAUTION:
The hypotheses should express the belief or suspicion we have before we see the data.

19. Stating Hypotheses AP® Exam Tip
CAUTION:
The hypotheses should express the belief or suspicion we have before we see the data.
AP® Exam Tip
Hypotheses always refer to a population, not to a sample. Be sure to state H0 and Ha in terms of population parameters. It is never correct to write a hypothesis about a sample statistic, such as H0: 𝑝 = or Ha: 𝑥 = 31.

20. Interpreting P-values
A player who claimed to make 80% of his free throws made only 𝑝 = =0.64 in a random sample of 50 free throws.

21. Interpreting P-values
A player who claimed to make 80% of his free throws made only 𝑝 = =0.64 in a random sample of 50 free throws.
This is evidence against the null hypothesis that p = 0.80 and in favor of the alternative hypothesis p < But is the evidence convincing?

22. Interpreting P-values
A player who claimed to make 80% of his free throws made only 𝑝 = =0.64 in a random sample of 50 free throws.
This is evidence against the null hypothesis that p = 0.80 and in favor of the alternative hypothesis p < But is the evidence convincing?
In other words, how likely is it for an 80% shooter to make 64% or less by chance alone in a random sample of 50 attempts?

23. Interpreting P-values
The P-value of a test is the probability of getting evidence for the alternative hypothesis Ha as strong or stronger than the observed evidence when the null hypothesis H0 is true.

24. Interpreting P-values
The P-value of a test is the probability of getting evidence for the alternative hypothesis Ha as strong or stronger than the observed evidence when the null hypothesis H0 is true.
P-value ≈ 3/400 =

25. Interpreting P-values
The P-value of a test is the probability of getting evidence for the alternative hypothesis Ha as strong or stronger than the observed evidence when the null hypothesis H0 is true.
P-value ≈ 3/400 =
We’ll show you how to calculate P-values later. For now, let’s focus on interpreting them.

26. Interpreting P-values
Problem: Calcium is a vital nutrient for healthy bones and teeth. The National Institutes of Health (NIH) recommends a calcium intake of 1300 milligrams (mg) per day for teenagers. The NIH is concerned that teenagers aren’t getting enough calcium, on average. Is this true? Researchers decide to perform a test of
H0: µ = 1300
Ha: µ < 1300
where µ is the true mean daily calcium intake in the population of teenagers. They ask a random sample of 20 teens to record their food and drink consumption for 1 day. The researchers then compute the calcium intake for each student. Data analysis reveals that 𝑥 =1198 mg and𝑠𝑥=411 mg. Researchers performed a significance test and obtained a P-value of
(a) Explain what it would mean for the null hypothesis to be true in this setting.
(b) Interpret the P-value.
Martin Shields/Alamy

27. Interpreting P-values
Problem: (a) Explain what it would mean for the null hypothesis to be true in this setting.
(b) Interpret the P-value.

28. Interpreting P-values
Problem: (a) Explain what it would mean for the null hypothesis to be true in this setting.
(b) Interpret the P-value.
(a) If H0: µ = 1300 is true, then the mean daily calcium intake in the population of teenagers is 1300 mg.
Martin Shields/Alamy

29. Interpreting P-values
Problem: (a) Explain what it would mean for the null hypothesis to be true in this setting.
(b) Interpret the P-value.
(a) If H0: µ = 1300 is true, then the mean daily calcium intake in the population of teenagers is 1300 mg.
(b) Assuming that the mean daily calcium intake in the teen population is 1300 mg, there is a probability of getting a sample mean of 1198 mg or less just by chance in a random sample of 20 teens.
Martin Shields/Alamy

30. Making Conclusions
We make a decision based on the strength of the evidence in favor of the alternative hypothesis (and against the null hypothesis) as measured by the P-value.

31. Making Conclusions
We make a decision based on the strength of the evidence in favor of the alternative hypothesis (and against the null hypothesis) as measured by the P-value.
If the observed result is unlikely to occur by chance alone when H0 is true (small P-value), we will “reject H0.”

32. Making Conclusions
We make a decision based on the strength of the evidence in favor of the alternative hypothesis (and against the null hypothesis) as measured by the P-value.
If the observed result is unlikely to occur by chance alone when H0 is true (small P-value), we will “reject H0.”
If the observed result is not unlikely to occur by chance alone when H0 is true (large P-value), we will “fail to reject H0.”

33. How to Make a Conclusion in a Significance Test
Making Conclusions
We make a decision based on the strength of the evidence in favor of the alternative hypothesis (and against the null hypothesis) as measured by the P-value.
If the observed result is unlikely to occur by chance alone when H0 is true (small P-value), we will “reject H0.”
If the observed result is not unlikely to occur by chance alone when H0 is true (large P-value), we will “fail to reject H0.”
How to Make a Conclusion in a Significance Test
If the P-value is small, reject H0 and conclude that there is convincing evidence for Ha (in context).
If the P-value is not small, fail to reject H0 and conclude that there is not convincing evidence for Ha (in context).

34. Making Conclusions
How small does a P-value have to be for us to reject H0?

35. Making Conclusions
How small does a P-value have to be for us to reject H0?
In Chapter 4 , we suggested that you use a boundary of 5% when determining whether a result is statistically significant.

36. Making Conclusions
How small does a P-value have to be for us to reject H0?
In Chapter 4 , we suggested that you use a boundary of 5% when determining whether a result is statistically significant.
That is equivalent to saying, “View a P-value less than 0.05 as small.”

37. Making Conclusions
How small does a P-value have to be for us to reject H0?
In Chapter 4 , we suggested that you use a boundary of 5% when determining whether a result is statistically significant.
That is equivalent to saying, “View a P-value less than 0.05 as small.”
Sometimes it may be preferable to use a different boundary value—like 0.01 or 0.10—when drawing a conclusion in a significance test.

38. Making Conclusions
The significance level α is the value that we use as a boundary for deciding whether an observed result is unlikely to happen by chance alone when the null hypothesis is true.

39. Making Conclusions
The significance level α is the value that we use as a boundary for deciding whether an observed result is unlikely to happen by chance alone when the null hypothesis is true.
If the P-value is less than α, we say that the result is “statistically significant at the α = ____ level.”

40. Making Conclusions
The significance level α is the value that we use as a boundary for deciding whether an observed result is unlikely to happen by chance alone when the null hypothesis is true.
If the P-value is less than α, we say that the result is “statistically significant at the α = ____ level.”
CAUTION:
α should be stated before the data are produced.

41. Making Conclusions
Problem: A company has developed a new deluxe AAA battery that is supposed to last longer than its regular AAA battery. However, these new batteries are more expensive to produce, so the company would like to be convinced that they really do last longer. Based on years of experience, the company knows that its regular AAA batteries last for 30 hours of continuous use, on average. The company selects an SRS of 15 deluxe AAA batteries and uses them continuously until they are completely drained. The sample mean lifetime is 𝑥 = hours. A significance test is performed using the hypotheses
H0: µ = 30
Ha: µ > 30
where µ is the true mean lifetime (in hours) of the deluxe AAA batteries. The resulting P-value is What conclusion would you make at the α = 0.05 level?

42. Making Conclusions Problem:
A significance test is performed using the hypotheses
H0: µ = 30
Ha: µ > 30
where µ is the true mean lifetime (in hours) of the deluxe AAA batteries. The resulting P-value is What conclusion would you make at the α = 0.05 level?

43. Making Conclusions Problem:
A significance test is performed using the hypotheses
H0: µ = 30
Ha: µ > 30
where µ is the true mean lifetime (in hours) of the deluxe AAA batteries. The resulting P-value is What conclusion would you make at the α = 0.05 level?
Because the P-value of > α = 0.05, we fail to reject H0. We don’t have convincing evidence that the true mean lifetime of the company’s deluxe AAA batteries is greater than 30 hours.

44. Making Conclusions Problem:
A significance test is performed using the hypotheses
H0: µ = 30
Ha: µ > 30
where µ is the true mean lifetime (in hours) of the deluxe AAA batteries. The resulting P-value is What conclusion would you make at the α = 0.05 level?
Never “accept H0” or conclude that H0 is true!
CAUTION:
Because the P-value of > α = 0.05, we fail to reject H0. We don’t have convincing evidence that the true mean lifetime of the company’s deluxe AAA batteries is greater than 30 hours.

45. Making Conclusions AP® Exam Tip
We recommend that you follow the two-sentence structure from the example when writing the conclusion to a significance test.
The first sentence should give a decision about the null hypothesis—reject H0 or fail to reject H0—based on an explicit comparison of the P-value to a stated significance level.
The second sentence should provide a statement about whether or not there is convincing evidence for Ha in the context of the problem.

46. Start here.
This is where we stopped yesterday.

47. Type I and Type II Errors
When we draw a conclusion from a significance test, we hope our conclusion will be correct. But sometimes it will be wrong.

48. Type I and Type II Errors
When we draw a conclusion from a significance test, we hope our conclusion will be correct. But sometimes it will be wrong.
A Type I error occurs if a test rejects H0 when H0 is true. That is, the test finds convincing evidence that Ha is true when it really isn’t.
A Type II error occurs if a test fails to reject H0 when Ha is true. That is, the test does not find convincing evidence that Ha is true when it really is.

49. Type I and Type II Errors
When we draw a conclusion from a significance test, we hope our conclusion will be correct. But sometimes it will be wrong.
A Type I error occurs if a test rejects H0 when H0 is true. That is, the test finds convincing evidence that Ha is true when it really isn’t.
A Type II error occurs if a test fails to reject H0 when Ha is true. That is, the test does not find convincing evidence that Ha is true when it really is.

50. Type I and Type II Errors
Problem: A potato chip producer and its main supplier agree that each shipment of potatoes must meet certain quality standards. If the producer determines that more than 8% of the potatoes in the shipment have “blemishes,” the truck will be sent away to get another load of potatoes from the supplier. Otherwise, the entire truckload will be used to make potato chips. To make the decision, a supervisor will inspect a random sample of 500 potatoes from the shipment. The producer will then perform a test at the α = 0.05 significance level of
H0: p = 0.08
Ha: p > 0.08
where p = the true proportion of potatoes with blemishes in a given truckload. Describe a Type I and a Type II error in this setting, and give a possible consequence of each.

51. Type I and Type II Errors
Problem: The producer will then perform a test at the α = 0.05 significance level of
H0: p = 0.08
Ha: p > 0.08
where p = the true proportion of potatoes with blemishes in a given truckload. Describe a Type I and a Type II error in this setting, and give a possible consequence of each.

52. Type I and Type II Errors
Problem: The producer will then perform a test at the α = 0.05 significance level of
H0: p = 0.08
Ha: p > 0.08
where p = the true proportion of potatoes with blemishes in a given truckload. Describe a Type I and a Type II error in this setting, and give a possible consequence of each.
Type I error: The producer finds convincing evidence that more than 8% of the potatoes in the shipment have blemishes, when the true proportion is really 0.08.

53. Type I and Type II Errors
Problem: The producer will then perform a test at the α = 0.05 significance level of
H0: p = 0.08
Ha: p > 0.08
where p = the true proportion of potatoes with blemishes in a given truckload. Describe a Type I and a Type II error in this setting, and give a possible consequence of each.
Type I error: The producer finds convincing evidence that more than 8% of the potatoes in the shipment have blemishes, when the true proportion is really 0.08.
Consequence: The potato-chip producer sends away the truckload of acceptable potatoes, wasting time and depriving the supplier of money.

54. Type I and Type II Errors
Problem: The producer will then perform a test at the α = 0.05 significance level of
H0: p = 0.08
Ha: p > 0.08
where p = the true proportion of potatoes with blemishes in a given truckload. Describe a Type I and a Type II error in this setting, and give a possible consequence of each.
Type II error: The producer does not find convincing evidence that more than 8% of the potatoes in the shipment have blemishes, when the true proportion is greater than 0.08.

55. Type I and Type II Errors
Problem: The producer will then perform a test at the α = 0.05 significance level of
H0: p = 0.08
Ha: p > 0.08
where p = the true proportion of potatoes with blemishes in a given truckload. Describe a Type I and a Type II error in this setting, and give a possible consequence of each.
Type II error: The producer does not find convincing evidence that more than 8% of the potatoes in the shipment have blemishes, when the true proportion is greater than 0.08.
Consequence: More potato chips are made with blemished potatoes, which may upset customers and lead to decreased sales.

56. Type I and Type II Errors
The most common significance levels are α = 0.05, α = 0.01, and α = 0.10.

57. Type I and Type II Errors
The most common significance levels are α = 0.05, α = 0.01, and α = 0.10.
Which one of these is the best choice for a given significance test? That depends on whether a Type I error or a Type II error is more serious.

58. Type I and Type II Errors
The most common significance levels are α = 0.05, α = 0.01, and α = 0.10.
Which one of these is the best choice for a given significance test? That depends on whether a Type I error or a Type II error is more serious.
Type I Error Probability
The probability of making a Type I error in a significance test is equal to the significance level α.

59. Type I and Type II Errors
We can decrease the probability of making a Type I error in a significance test
by using a smaller significance level.

60. Type I and Type II Errors
We can decrease the probability of making a Type I error in a significance test
by using a smaller significance level.
But there is a trade-off between P(Type I error) and P(Type II error): as one increases, the other decreases.

61. Type I and Type II Errors
We can decrease the probability of making a Type I error in a significance test
by using a smaller significance level.
But there is a trade-off between P(Type I error) and P(Type II error): as one increases, the other decreases.
If we make it more difficult to reject H0 by decreasing α, we increase the probability that we will not find convincing evidence for Ha when it is true.

62. Section Summary
STATE appropriate hypotheses for a significance test about a population parameter.
INTERPRET a P-value in context.
MAKE an appropriate conclusion for a significance test.
INTERPRET a Type I error and a Type II error in context. GIVE a consequence of each error in a given setting.

63. Assignment 9.1 p. 565-567 #22-28 even 29-33 all
If you are stuck on any of these, look at the odd before or after and the answer in the back of your book. If you are still not sure text a friend or me for help (before 8pm).
Tomorrow we will check homework and review for 9.1 Quiz.