1. AP STATISTICS Summer Institute 2016 Day 3
Lance Belin
JJ Pearce High School – Richardson, TX Years
AP Statistics Reader/Table Leader – College Board Years
Masters in Statistics - University of Texas - Dallas

2. APSI schedule: August 4, 2016
8:30 AM to 4:00 PM
Morning Break: 10:15 AM
Lunch: 12: :30
Afternoon Break: 2:15 PM

6. This course covers the following topics:
Exploring Data: Describing patterns and departures from patterns (20%–30%)
Exploratory analysis of data makes use of graphical and numerical techniques to study patterns and departures from patterns. Emphasis should be placed on interpreting information from graphical and numerical displays and summaries.
Constructing and interpreting graphical displays of distributions of univariate data (dotplot, stemplot, histogram, cumulative frequency plot)
Summarizing distributions of univariate data
Comparing distributions of univariate data (dotplots, back-to-back stemplots, parallel boxplots)
Exploring bivariate data
Exploring categorical data

7. Sampling and Experimentation: Planning and conducting a study (10%–15%)
Data must be collected according to a well-developed plan if valid information on a conjecture is to be obtained. This plan includes clarifying the question and deciding upon a method of data collection and analysis.
Overview of methods of data collection
Planning and conducting surveys
Planning and conducting experiments
Generalizability of results and types of conclusions that can be drawn from observational studies, experiments, and surveys
Anticipating Patterns: Exploring random phenomena using probability and simulation (20%–30%)
Probability is the tool used for anticipating what the distribution of data should look like under a given model.
Probability
Combining independent random variables
The normal distribution
Sampling distributions

8. IV. Statistical Inference: Estimating population parameters and testing hypotheses (30%–40%)
Statistical inference guides the selection of appropriate models.
Estimation (point estimators and confidence intervals) Mean, Proportion, Slope
Estimating population parameters and margins of error
Tests of significance for a proportion, mean, slope
Logic of significance testing, null and alternative hypotheses; p-values; one- and two-sided tests; concepts of Type I and Type II errors; concept of power
Sample test Chi Square

10. https://apstudent.collegeboard.org/apcourse/ap-statistics

11. An Exercise in Sampling: Rolling Down the River Page 22-29
A farmer has just cleared a new field for corn. It is a unique plot of land in that a river runs along one side. The field is harder to harvest the closer you get to the river. By surveying his land, the farmer notices that the corn looks good in some areas of the field but not others. The farmer is not sure that harvesting the field is worth the expense. He will harvest the field if he believes that there is at least an average of 75 bushels per acre per plot. He has decided to harvest 10 plots out of the 100 total plots and use this information to estimate the total yield. Based on this estimate, he will decide whether to harvest the remaining plots. His property is shown below.

12. An Exercise in Sampling: Rolling Down the River Page 22-29
Convenience Sample
Simple Random Sample
Stratified Sample
Cluster Sample

13. Goldfish Page 10  DO NOT EAT!!!!!!
Purpose: To simulate counting large populations using the Central Limit Theorem and confidence intervals (LEVELS).
1. After being given your materials, count the number of tagged fish. (pretzel)
2. Add the tagged fish to your pond.
3. As a group, catch a random sample of fish in your net, recording the proportion of tagged fish to the total number of fish. Use the entire size of your net.
4. If we knew the true proportion of tagged fish, p, what would be ?
5. We don’t know p, but we do know p-hat from step 3. What is true about p and , especially is the sample size was large? (Therefore, the answer from above will be close to )

15. 6 -7. Calculate the 90% Confidence Interval
8. Take turns repeating this process until you have a total of 20 such intervals. Each trial does not need to have the same number of fish, but should be relatively large and close to the same size.
9. Graph all 20 intervals using the graph paper provided. The scale on the x-axis is from – .5 to 1.5 with a scale of 0.1 and the y-axis should be used for the trial numbers, stacking the confidence intervals on top of each other.
10 ?Tag fish
11.

16. 2015 AP STATISTICS EXAM Question #2 Page 12
To increase business, the owner of a restaurant is running a promotion in which a customer’s bill can be randomly selected to receive a discount. When a customer’s bill is printed, a program in the cash register randomly determines whether the customer will receive a discount on the bill. The program was written to generate a discount with a probability of 0.2, that is, giving 20 percent of the bills a discount in the long run. However, the owner is concerned that the program has a mistake that results in the program not generating the intended long-run proportion of 0.2.
The owner selected a random sample of bills and found that only 15 percent of them received discounts. A confidence interval for p, the proportion of bills that will receive a discount in the long run, is ±
All conditions for inference were met.
(a)Consider the confidence interval 0.15 ±
(i) Does the confidence interval provide convincing statistical evidence that the program is not working as intended? Justify your answer.
(ii) Does the confidence interval provide convincing statistical evidence that the program generates the discount with a probability of 0.2 ? Justify your answer.

17. 2015 AP STATISTICS EXAM Question #2 Page 12
(a)Consider the confidence interval 0.15 ±
(i) Does the confidence interval provide convincing statistical evidence that the program is not working as intended? Justify your answer.
(ii) Does the confidence interval provide convincing statistical evidence that the program generates the discount with a probability of 0.2 ? Justify your answer.

18. 2015 AP STATISTICS EXAM Question #2 Page 12
A second random sample of bills was taken that was four times the size of the original sample. In the second sample 15 percent of the bills received the discount.
(b) Determine the value of the margin of error based on the second sample of bills that would be used to compute an interval for p with the same confidence level as that of the original interval.
(c) Based on the margin of error in part (b) that was obtained from the second sample, what do you conclude about whether the program is working as intended? Justify your answer.

19. 2016 AP STATISTICS EXAM Question #5

20. Student Response 5A1

21. Student Response 5A2 (b)

22. Student Response 5A2 (c)

23. Student Response 5D

24. Student Response 5F

25. Student Response 5I

26. Andy scored a 650 on the Math portion of the SAT
Andy scored a 650 on the Math portion of the SAT. Mary scored a 25 on the ACT. If the Math SAT has a Normal Distribution, , and the ACT has a Normal Distribution , which student did better and why?
ACT
SAT
Andy = (650 – 500 )/95 = 1.58
Mary = (25 – 21)/3 = 1.33

27. Probability & Paper Wads Page 53

28. Made shot Missed Shot Total Males Females Data Collection Page 53
Put on board

29. a) What is the probability that a shot was made?
P(shot made) =
b) What is the probability that a shot was taken by a female?
P(female) =

30. c) What is the probability a shot was made
and the shooter was a female?
P(shot made and female) =
d) What is the probability a shot was made
or the shooter was a female?
P(shot made or female) =

31. e) Given that a shot was made, what is the probability the shooter was a male?
P(male | shot made) =
f) Are making the shot and being male independent? Justify your answer.
Test for independence: P(A) P(A | B)

32. g) Assuming the probabilities above remain true for future shots, what is the probability two shots in a row both land in the wastebasket?
P(2 shots made) =

33. 2015 AP STATISTICS EXAM Question #3

34. 2015 AP STATISTICS EXAM Question #3

35. PROBABILITY EXAMPLES Remember:
Check to see if the data are categorical or quantitative (means or proportions?)
Find the mean and standard deviation of the sampling distribution….if needed.
Check the conditions
Sketch the curve and show your work towards the solution.
Write your answer in proper form.
Answer the question.
Pair up and randomly call students to the board.

36. The proportion of adventurous hobbits in a traditional hobbit community is 8%. Gandalf randomly selected 200 hobbits from a remote forest community and found 12.5% willing to time-travel to the twenty-first century. Could this hairy footed community really be called traditional?
No, the probability of randomly selecting 200 hobbits with 12.5% willing to time travel is too low to have happened by chance (0.009)

37. Suppose that a candidate for public office is favored by 45% of all registered voters in the district. A group of statistics students calls a random sample of 500 voters to assess the candidate’s standing. What is the approximate probability that will be greater than .5, causing the students to incorrectly predict a win for the candidate?
(So in other words, out of all the possible s, what proportion of them will be greater than .5?)
0.0123

38. The ages of college students at a large university have a mean of 25 years and a standard deviation of 3 years. Find the probability that the mean age for a random sample of 36 students would be less than 24 years.
0.0228

39. Suppose the proportion of bats that live under the Congress Street bridge that have rabies is 15%. If 70 bats are randomly selected and tested for rabies, what is the probability that the proportion of bats in the sample that test positive for rabies is more than 25%?
0.0096

40. Suppose it is known that the increased temperature of the heated water discharged by a certain power plant on any given day has a distribution with a mean of 5˚C and a standard deviation of 0.5˚C. For 50 randomly selected days, what is the approximate probability that the average increase in temperature of the discharged water is greater than 5.2˚C?
.389

41. The zombie apocalypse has arrived, to the surprise of some
The zombie apocalypse has arrived, to the surprise of some. During the beginning of the apocalypse, the proportion of zombies in Texas is In a random sample of 32 Richardson graduates, what is the probability that at least 75% will be mindless flesheaters?
.1977

42. According to government data, 22% of American children under the age of 6 live in households with incomes less than the official poverty level. A random sample of 300 children is selected for a study of learning in early childhood. What is the probability that 80 or more of the children in the sample live in poverty?
.0246

43. According to government data, 22% of American children under the age of 6 live in households with incomes less than the official poverty level. A random sample of 20 children is selected for a study of learning in early childhood.
a.) What is the probability that 8 or more of the children in the sample live in poverty?
b.) What is the probability that 8 of the children in the sample live in poverty?
.0536
.0351
normal or binomial method? Try both if time.

44. A hotdog manufacturer claims that one of its brands of hot dogs has an average fat content of 18 g per dog. Consumers of this brand would probably not mind if the mean is less than 18 g but would be upset if it exceeds 18. Suppose the standard deviation is one gram of fat.
An independent testing organization is asked to analyze a random sample of 36 hot dogs, and find that their sample mean, , is 18.4 g of fat. Does this suggest that the manufacturer’s claim is incorrect?

45. The Harvard College Alcohol Study finds that 67% of college students sampled nationally (n=15000) support the efforts to “crack down on underage drinking.” Upon reading the study, a local college takes a random sample of 100 students and finds that 62% of it’s students support a crackdown on underage drinking. The campus newspaper then writes an article claiming that “support at our campus is lower at our school than nationally.” Does the survey support that claim?

46. Page 46 Has the color distribution of M&M’s changed
Page 46 Has the color distribution of M&M’s changed? (based on an Activity in TPS 5e) According to the M&M’s website in 2009, the milk chocolate variety of M&M’s has the following color distribution: Red: 13% Brown: 13% Yellow: 14% Green: 16% Orange: 20% Blue: 24%

47. 2015 AP STATISTICS EXAM Question #4S

48. 2015 AP STATISTICS EXAM Question #4SH PH

49. 2015 AP STATISTICS EXAM Question #4S

50. 2015 AP STATISTICS EXAM Question #4SMS

51. 2016 AP STATISTICS EXAM Question #2

52. Student Sample 2A1

53. Student Sample 2A1

54. Student Sample 2A1

55. Student Sample 2B

56. Student Sample 2B

57. Student Sample 2 I

58. Introducing Inference and Alpha using Cards
First method:
Second method: Page 19