1. Confidence Intervals for Proportions
Ch. 19 Notes
AP Statistics

2. Chapter 19 Textbook HW
p.455 #2,4,6,13 (Goal for Tonight) #7,8,14,18,22,26,30,32,35,36

3. P. 443 Graphs

4. Inference
To infer means to draw a conclusion.

5. Confidence Level
The success rate of the method used to construct the interval.
It provides information on how much “confidence” we can have in the method used to construct the interval estimate.

6. Confidence Level Example
We have used a method to produce this estimate that is successful in capturing the actual population proportion 90% of the time

7. Confidence Interval
It is an interval computed from sample data that contains the plausible values for the characteristic being studied.
It is constructed so that, with a chosen degree of confidence, the actual value of the population characteristic will be between the lower and upper endpoints of the interval.

8. Confidence Interval
Estimate +/- margin of error

9. Margin of Error
Shows how accurate we believe our guess is based on the variability of the estimate
Only covers random sampling errors

11. CI Behavior Confidence level increases-interval width increases
σ decreases-interval width decreases
Sample size increases-interval width decreases

12. Interpretation Practice

13. JC P. 442
Pew asked cell phone owners, “Have you ever received unsolicited text messages on your cell phone from advertisers?” and 17% reported that they had. Pew estimates a 95% confidence interval to be 0.17 ± 0.04, or between 13% and 21%.

14. JC P. 442
In Pew’s sample, somewhere between 13% and 21% of respondents reported that they had received unsolicited advertising text messages.
We can by 95% confident that 17% of all US cell phone owners have received unsolicited advertising text messages.

15. We are 95% confident that between 13% and 21% of all US cell phone owners have received unsolicited text messages.
We know that between 13% and 21 of all US cell phone owners have received unsolicited text messages.
95% of all US cell phone owners have received unsolicited text messages.

16. JC P. 445
On January 30-31, 2007, Fox News polled 900 registered voters nationwide. When asked “Do you believe global warming exists?” 82% said yes. Fox reported their margin of error to be ± 3%.Think some more about the 95% confidence interval Fox News created for the proportion of registered voters who believe that global warming exists.

17. JC P. 445
If Fox wanted to be 98% confident would their CI need to be wider or narrower?
Fox’s ME was about 3%. If they reduced it to 2.5%, would their level of confidence by higher or lower?
If Fox News had polled more people, would the ME be larger or smaller?

18. Example P. 443
Fox News asked 900 registered voters nationwide, “Do you believe global warming exists?” 82% said yes. Fox reported their margin of error to be ±3%. It is standard among pollsters to use a 95% confidence level unless otherwise stated. Given that, what does Fox News mean by claiming a margin of error of ±3% in this context?

19. Standard Error
𝑆𝐸 𝑝 = 𝑝 𝑞 𝑛 This is the estimate of the standard deviation of a sampling distribution.

20. Margin of Error formula
𝑆𝐸 𝑝 = 𝑝 𝑞 𝑛
𝑧 ∗ ×
To use InvNorm: take confidence level as decimal + 1 and then ÷ 2 and calculate using InvNorm with
Mean = 0 and SD = 1
Critical value- depends on confidence level- use chart or can use InvNorm

21. Confidence Interval formula
𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒±𝑚𝑎𝑟𝑔𝑖𝑛 𝑜𝑓 𝑒𝑟𝑟𝑜𝑟
𝑝 ±𝑧* 𝑝 𝑞 𝑛

22. Example P. 445
A Fox News poll of 900 registered voters found that 82% of the respondents believed that global warming exists. Fox reported a 95% CI with a ME of 3%.
What would the margin of error for a 90% confidence interval be?
What is good and bad about this change?

23. Conditions/Assumptions
Independence
Randomization
10%/Sample Size
Success/Failure

24. Steps Conditions 1a. Picture Formula Plug numbers into formula
Conclusion/disclaimer

25. Conclusion Statement Confidence Interval
We are _____% confident that the true population proportion of “context” falls between ____ and _______.

26. Confidence Level Statement
_____% of the intervals constructed using this method will include the proportion of context.

27. Disclaimer
Since all of the conditions are not met, the results may not be accurate.

28. Chapter 19 Confidence Intervals for Proportions Day 2

29. In January 2007 Consumer Reports published their study of bacterial contamination of chicken sold in the US. They purchased 525 broiler chickens from various kinds of food stores in 23 states and tested them for types of bacteria that cause food borne illnesses. Lab results indicated that 83% of these chickens were infected with Campylobacter.
Example 1

30. Construct a 90% confidence interval.

31. An insurance company checks police records on 582 accidents selected at random and notes that teenagers were at the wheel in 91 of them.
Create a 95% confidence interval.
B. Explain what your interval means.
Example 2

32. C. A politician urging tighter restrictions on drivers’ licenses issued to teens says, “In one of every five auto accidents, a teenager is behind the wheel.” Does your confidence interval support or contradict this statement? Explain.
Example 2 (cont)

33. Trash Ball
Each person will try to make a basket in the garbage can. Each person will toss their paper ball from a specified location in the room and record if they make it or not. We will then construct a 96% confidence interval for the proportion of baskets made into the garbage can.
Example 3

34. Sample Size (solve for n)
𝑀𝐸=𝑧∗ 𝑝 𝑞 𝑛

35. The Fox News poll which estimated that 82% of all voters believed global warming exists had a margin of error of 3%. Suppose an environmental group planning a follow up survey of voters’ opinions on global warming wants to determine a 95% confidence interval with a margin of error of no more than 2%. How large a sample do they need?
Example 4

36. A credit card company is about to send out a mailing to test the market for a new credit card. From that sample, they want to estimate the true proportion of people who will sign up for the card nationwide. A pilot study suggests that about 0.5% of the people receiving the offer will accept it. To be within a tenth of a percentage point(0.001) of the true rate with 95% confidence, how big does the test mailing have to be?
Example 5

37. Be Careful!
Don’t suggest that the parameter varies. Make it clear you know the parameter is fixed and the interval changes from sample to sample.
Don’t claim that other samples will agree with yours.
Don’t be certain about the parameter. We can’t be absolutely certain of anything!

38. Be Careful!
Don’t forget. It’s the parameter. The confidence interval is about the unknown parameter.
Don’t claim to know too much. Do not use the word all.
Do take responsibility. You have to accept the fact that not all intervals will capture the true value.

39. Be Careful!
Watch out for biased sampling.
Think about independence.

40. Practice AP Question
Year: 2000 #6

41. A random sample of 400 married couples was selected from a large population of married couples.
Heights of married men are approximately normally distributed with a mean 70 inches and standard deviation 3 inches.
Heights of married women are approximately normally distributed with mean 65 inches and standard deviation 2.5 inches.
There were 20 couples in which the wife was taller than her husband, and there were 380 couples in which the wife was shorter than her husband.
Find a 95 percent confidence interval for the proportion of married couples in the population for which the wife is taller than her husband. Interpret your interval in the context of this question.
Warm Up

42. Solution (2000 – Question 6, part a)
Parameter, Assumptions, Test Name (or formula):
p = true proportion of married couples in which the
wife is taller than her husband
Assumption: large sample size since
1-proportion z-interval for p

43. Solution (2000 – Question 6, part a)
Calculations:

44. Solution (2000 – Question 6, part a)
Interpret the interval:
I am 95% confident that the true proportion of couples in which the wife is taller than her husband is between .028 and .071, based on this sample.

45. Tooth Fairy
The lives of young children growing up in American are complete with many figures, some real, and some slightly less so, such as the Tooth Fairy. Researchers in Michigan were interested in the strength and duration of such beliefs as characteristics of the child’s psychological and cognitive development. Specifically they were interested in the ages at which beliefs in such fantasy figures declined.

46. The investigators interviewed white, middle class, Christian children in southeastern Michigan, and categorized them as “firm believers” in various fantasy figures. The data is on the next slide. Our task will be to construct a 99% CI for the population proportion of firm believers in each age group.

47. Age(years)
Firm Believers
Not firm believers
6-7 29 21
8-10 12 35