1. Section 7.1
Review and Preview

2. Review
Chapters 2 & 3 we used “descriptive statistics” when we summarized data using tools such as graphs, and statistics such as the mean and standard deviation.
Chapter 6 we introduced critical values: z denotes the z score with an area of  to its right.
If  = 0.025, the critical value is z0.025 = That is, the critical value z0.025 = 1.96 has an area of to its right.

3. Preview This chapter presents the beginning of inferential statistics.
The two major activities of inferential statistics are (1) to use sample data to estimate values of a population parameters, and (2) to test hypotheses or claims made about population parameters.
We introduce methods for estimating values of these important population parameters: proportions, means, and variances.
We also present methods for determining sample sizes necessary to estimate those parameters.

4. Estimating a Population Proportion
Section 7.2
Estimating a Population Proportion

5. Learning Targets: In this section….
We present methods for using a sample proportion to estimate the value of a population proportion.
The sample proportion is the best point estimate of the population proportion.
We can use a sample proportion to construct a confidence interval to estimate the true value of a population proportion, and we should know how to interpret such confidence intervals.
We should know how to find the sample size necessary to estimate a population proportion.

6. A point estimator is a statistic that provides an estimate of a population parameter.
Point estimators that we will be using are
The value of that statistic from a sample is called a point estimate. Ideally, a point estimate is our “best guess” at the value of an unknown parameter.
The sample proportion is the best point estimate of the population proportion p.
The sample mean is the best point estimate of the population mean µ.

7. Example 1: Determine the point estimator you would use and calculate the value of the point estimate.
a) Quality control inspectors want to estimate the mean lifetime μ of the AA batteries produced in an hour at a factory. They select a random sample of 30 batteries during each hour of production and then drain them under conditions that mimic normal use. Here are the lifetimes (in hours) of the batteries from one such sample:

8. b) What proportion, p, of U. S. high school students smoke
b) What proportion, p, of U.S. high school students smoke? The 2007 Youth Risk Behavioral Survey questioned a random sample of 14,041 students in grades 9 to 12. Of these, 2808 said they had smoked cigarettes at least one day in the past month.

9. Example 2: A Pew Research Center poll, 70% of 1501 randomly selected adults in the United States believe in global warming, so the sample proportion is = Find the best point estimate of the proportion of all adults in the United States who believe in global warming.

10. A confidence interval (or interval estimate) is a range (or an interval) of values used to estimate the true value of a population parameter. A confidence interval is sometimes abbreviated as CI.

11. Most common choices are 90%, 95%, or 99%.
A confidence level is the probability 1 –  (often expressed as the equivalent percentage value) that the confidence interval actually does contain the population parameter, assuming that the estimation process is repeated a large number of times. (The confidence level is also called degree of confidence, or the confidence coefficient.)
Most common choices are 90%, 95%, or 99%.
(α = 10%), (α = 5%), (α = 1%)

12. Interpreting a Confidence Interval
We must be careful to interpret confidence intervals correctly. There is a correct interpretation and many different and creative incorrect interpretations of the confidence interval < p <
“We are 95% confident that the interval from to actually does contain the true proportion of [the topic].”
This means that if we were to select many different samples of size 1501 (from example 2) and construct the corresponding confidence intervals, 95% of them would actually contain the value of the population proportion p.
(Note that in this correct interpretation, the level of 95% refers to the success rate of the process being used to estimate the proportion.)

13. Example 3: The 96% confidence interval for the true
proportion of all 17 year old boys who own a used car with a
sample size of 426 is (0.189, 0.251). Interpret this confidence
interval.
We are 96% confident that the interval from to actually does contain the true proportion of all 17 year old boys who own a used car.

14. Example 4: The 95% confidence interval for the true
proportion of all New York State Union members who favor
the Republican candidate for governor with a sample size of
300 is (0.147, 0.321). Interpret this confidence interval.
We are 95% confident that the interval from to actually does contain the true proportion of all New York State Union members who favor the Republican candidate for governor.

15. Critical Values
A standard z score can be used to distinguish between sample statistics that are likely to occur and those that are unlikely to occur. Such a z score is called a critical value. Critical values are based on the following observations:
1. Under certain conditions, the sampling distribution of sample proportions can be approximated by a normal distribution.

16. Critical Values
2. A z score associated with a sample proportion has a probability of 𝜶 𝟐 of falling in the right tail.

17. Critical Values
3. The z score separating the right-tail region is commonly denoted by z/2 (z* (“z star”) and is referred to as a critical value because it is on the borderline separating z scores from sample proportions that are likely to occur from those that are unlikely to occur.

18. Notation for Critical Value
The critical value z/2 (z*) is the positive z value that is at the vertical boundary separating an area of /2 in the right tail of the standard normal distribution. (The value of –z/2 is at the vertical boundary for the area of /2 in the left tail.) The subscript /2 is simply a reminder that the z score separates an area of /2 in the right tail of the standard normal distribution.

19. Finding zα/2 (z*) for a 95% Confidence Level
α = 5%
α/2 = 2.5% = .025
-zα/2
zα/2
Critical Values

20. Finding z2 (z*) for a 95% Confidence Level – cont.
α = 0.05
zα/2 = ±1.96

21. Example 5: Find the appropriate critical value for the given
confidence level. Round to three decimals.
a) 99.9%
b) 95%
c) 90%

22. Example 5 cont.: Find the appropriate critical value for the
given confidence level. Round to three decimals.
d) 92%
e) 84%
f) 78%

23. Margin of Error for Proportions
When data from a simple random sample are used to estimate a population proportion p, the margin of error, denoted by E, is the maximum likely difference (with probability 1 – , such as 0.95) between the observed proportion and the true value of the population proportion p.
Margin of Error for Proportions

24. Example 6: Assume that a sample is used to estimate a
population proportion p. Find the margin of error, E, that
corresponds to the given statistics and confidence level.
Round the margin of error to three decimal places.
a) 95% confidence, sample size is 600, of 32% are successes.
b) 98% confidence, sample size is 1142, of 24% are successes.

25. Example 7: In a random sample of 203 college students, 75
had part-time jobs. Find the margin of error for the 90%
confidence interval used to estimate the population proportion.

26. Confidence Interval for Estimating a Population Proportion p
p = population proportion
= sample proportion
n = number of sample values
E = margin of error
z/2 (z*) = z score separating an area of /2 in the
right tail of the standard normal
distribution

27. Confidence Interval for Estimating a Population Proportion p
where

28. 3 Different ways for Writing Confidence Interval for a Population Proportion p

29. Example 8: A Pew Research Center poll of 1501 randomly selected U. S
Example 8: A Pew Research Center poll of 1501 randomly selected U.S. adults showed that 70% of the respondents believe in global warming. The sample results are n = 1501 and
a) Find the margin of error E that corresponds to a 95% confidence level.
b) Find the 95% confidence interval estimate of the population proportion p.

30. Calculator Setup 1st: On your graphing calculator, go to STAT  TESTS
2nd: Scroll Up, then Choose A:
1-PropZInt…

31. Example 9: Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p.
n = sample size, x = number of successes
a) n = 741, x = 274; 95% confidence
b) n = 267, x = 194; 88% confidence

32. Sample Size
Suppose we want to collect sample data in order to estimate some population proportion. The question is how many sample items must be obtained?

33. Determining Sample Size
(solve for n by algebra)

34. Sample Size for Estimating Proportion p
When an estimate of is known:
When an estimate of is unknown:

35. Round-Off Rule for Determining Sample Size
If the computed sample size n is not a whole number, round the value of n up to the next larger whole number.

36. Example 10: The Internet is affecting us all in many different ways, so there are many reasons for estimating the proportion of adults who use it. Assume that a manager for E-Bay wants to determine the current percentage of U.S. adults who now use the Internet. How many adults must be surveyed in order to be 95% confident that the sample percentage is in error by no more than three percentage points?
a) In 2006, 73% of adults used the Internet.
b) No known possible value of the proportion.

37. Example 11: Use the given data to find the minimum sample size required to estimate the population proportion.
a) Margin of error: 0.006; confidence level: 90%; unknown.
b) Margin of error: 0.02; confidence level: 94%; unknown.

38. Example 12: Use the given data to find the minimum sample size required to estimate the population proportion.
a) Margin of error: 0.08; confidence level: 96%; from a prior study, is estimated by 0.24.
b) Margin of error: 0.004; confidence level: 92%; from a prior study, is estimated by

39. Example 13: The Genetics and IVF Institute conducted a clinical trial of the
XSORT method designed to increase the probability of conceiving a girl. As of
this writing, 574 babies were born to parents using the XSORT method, and 525
of them were girls.
a) What is the best point estimate of the population proportion of girls born to
parents using the XSORT method?
b) Use the sample data to construct a 95% confidence interval estimate of the
percentage of girls born to parents using the XSORT method.

40. Example 14: An interesting and popular hypothesis is that individuals can
temporarily postpone their death to survive a major holiday or important event
Such as a birthday. In a study of this phenomenon, it was found that in the week
before and the week after Thanksgiving, there were 12,000 total deaths, and 6,062
of them occurred in the week before Thanksgiving (based on data from “Holiday,
Birthdays, and Postponement of Cancer Death,” by Young and Hade, Journal of
the American Medical Association, Vol. 292, No. 24.)
a) What is the best point estimate of the proportion of deaths in the week before
Thanksgiving to the total deaths in the week before and the week after Thanks–
giving?
b) Construct a 95% confidence interval estimate of the proportion of deaths in the
week before Thanksgiving to the total deaths in the week before and the week
after Thanksgiving?

41. Finding the Point Estimate and E from a Confidence Interval
When you already know the interval and are looking for or E

42. Example 15: The following confidence interval is obtained for a population proportion, p: (0.426, 0.612). Use these confidence interval limits to find the point estimate, .

43. Example 16: The following confidence interval is obtained for a population proportion, p: < p < Use these confidence interval limits to find the point estimate, .

44. Example 17: The following confidence interval is obtained for a population proportion, p: (0.647, 0.875). Use these confidence interval limits to find the margin of error, E.

45. Example 18: The following confidence interval is obtained for a population proportion, p: < p < Use these confidence interval limits to find the margin of error, E.

46. Example 19: Express the confidence interval 0. 052 < p < 0
Example 19: Express the confidence interval < p < in the form

47. Example 20: Express the confidence interval (0. 258, 0
Example 20: Express the confidence interval (0.258, 0.789) in the form