1. Chapter 4 Simple Random Sampling
Definition of Simple Random Sample (SRS) and how to select a SRS
Estimation of population mean and total; sample size for estimating population mean and total
Estimation of population proportion; sample size for estimating population proportion
Comparing estimates

2. Simple Random Samples
Desire the sample to be representative of the population from which the sample is selected
Each individual in the population should have an equal chance to be selected
Is this good enough?

3. Example Select a sample of high school students as follows:
Flip a fair coin
If heads, select all female students in the school as the sample
If tails, select all male students in the school as the sample
Each student has an equal chance to be in the sample
Every sample a single gender, not representative
Each individual in the population has an equal chance to be selected. Is this good enough?
NO!!

4. Simple Random Sample
A simple random sample (SRS) of size n consists of n units from the population chosen in such a way that every set of n units has an equal chance to be the sample actually selected.

5. Simple Random Samples (cont.)
Suppose a large History class of 500 students has 250 male and 250 female students.
To select a random sample of 250 students from the class, I flip a fair coin one time.
If the coin shows heads, I select the 250 males as my sample; if the coin shows tails I select the 250 females as my sample.
What is the chance any individual student from the class is included in the sample?
This is a random sample. Is it a simple random sample?
1/2
NO!
Not every possible group of 250 students has an equal chance to be selected.
Every sample consists of only 1 gender – hardly representative.

6. Simple Random Samples (cont.)
The easiest way to choose an SRS is with random numbers.
Statistical software can generate random digits (e.g., Excel “=random()”,
ran# button on
calculator).

7. Example: simple random sample
Academic dept wishes to randomly choose a 3-member committee from the 28 members of the dept
00 Abbott 07 Goodwin 14 Pillotte 21 Theobald
01 Cicirelli 08 Haglund 15 Raman 22 Vader
02 Crane 09 Johnson 16 Reimann 23 Wang
03 Dunsmore 10 Keegan 17 Rodriguez 24 Wieczoreck
04 Engle 11 Lechtenb’g 18 Rowe 25 Williams
05 Fitzpat’k 12 Martinez 19 Sommers 26 Wilson
06 Garcia 13 Nguyen 20 Stone 27 Zink

8. Solution
Use a random number table; read 2-digit pairs until you have chosen 3 committee members
For example, start in row 121:
Garcia (07) Theobald (22) Johnson (10)
Your calculator generates random numbers; you can also generate random numbers using Excel

9. Sampling Variability Suppose we had started in line 145?
Our sample would have been
19 Rowe, 26 Williams, 06 Fitzpatrick

10. Variability is OK; bias is bad!!
Sampling Variability
Samples drawn at random generally differ from one another.
Each draw of random numbers selects different people for our sample.
These differences lead to different values for the variables we measure.
We call these sample-to-sample differences sampling variability.
Variability is OK; bias is bad!!

11. Example: simple random sample
Using Excel tools
Using statcrunch (NFL)

12. 4.3 Estimation of population mean 
Usual estimator

13. 4.3 Estimation of population mean 
For a simple random sample of size n chosen without replacement from a population of size N
The correction factor takes into account that an estimate based on a sample of n=10 from a population of N=20 items contains more information than a sample of n=10 from a population of N=20,000

14. 4.3 Estimating the variance of the sample mean
Recall the sample variance

15. 4.3 Estimating the variance of the sample mean

16. 4.3 Estimating the variance of the sample mean

17. 4.3 Example Population {1, 2, 3, 4}; n = 2, equal weights Sample
Pr. of sample s2
{1, 2}
1/6
1.5
0.5
0.125
{1, 3}
2.0
0.500
{1, 4}
2.5
4.5
1.125
{2, 3}
{2, 4}
3.0
{3, 4}
3.5

18. 4.3 Example
Population {1, 2, 3, 4}; =2.5, 2 = 5/4; n = 2, equal weights
Sample
Pr. of sample s2
{1, 2}
1/6
1.5
0.5
0.125
{1, 3}
2.0
0.500
{1, 4}
2.5
4.5
1.125
{2, 3}
{2, 4}
3.0
{3, 4}
3.5

19. 4.3 Example
Population {1, 2, 3, 4}; =2.5, 2 = 5/4; n = 2, equal weights
Sample
Pr. of sample s2
{1, 2}
1/6
1.5
0.5
0.125
{1, 3}
2.0
0.500
{1, 4}
2.5
4.5
1.125
{2, 3}
{2, 4}
3.0
{3, 4}
3.5

20. 4.3 Example Summary
Population {1, 2, 3, 4}; =2.5, 2 = 5/4; n = 2, equal weights
Sample
Pr. of sample s2
{1, 2}
1/6
1.5
0.5
0.125
{1, 3}
2.0
0.500
{1, 4}
2.5
4.5
1.125
{2, 3}
{2, 4}
3.0
{3, 4}
3.5

21. 4.3 Margin of error when estimating the population mean 

22. t distributions Very similar to z~N(0, 1)
Sometimes called Student’s t distribution; Gossett, brewery employee
Properties:
i) symmetric around 0 (like z)
ii) degrees of freedom 

23. Student’s t Distribution
P(t < ) = .025
P(t > ) = .025
.95
.025
.025
t10
2.2281

24. Standard normal P(z < -1.96) = .025 P(z > 1.96) = .025 .95 .025
1.96

25. Student’s t Distribution-3 -2 -1 1 2 3 Z t
Figure 11.3, Page 372

26. Student’s t Distribution
Degrees of Freedom -3 -2 -1 1 2 3 Z t1
Figure 11.3, Page 372

27. Student’s t Distribution
Degrees of Freedom -3 -2 -1 1 2 3 Z t1 t7
Figure 11.3, Page 372

28. 4.3 Margin of error when estimating the population mean 

29. 4.3 Margin of error when estimating the population mean 
Understanding confidence intervals; behavior of confidence intervals.

30. 4.3 Margin of error when estimating the population mean 

31. Comparing t and z Critical Values
Conf.
level n = 30
z = % t =
z = % t =
z = % t =
z = % t =

32. 4.4 Determining Sample Size to Estimate 

33. Required Sample Size To Estimate a Population Mean 
If you desire a C% confidence interval for a population mean  with an accuracy specified by you, how large does the sample size need to be?
We will denote the accuracy by MOE, which stands for Margin of Error.

34. Example: Sample Size to Estimate a Population Mean 
Suppose we want to estimate the unknown mean height  of male students at NC State with a confidence interval.
We want to be 95% confident that our estimate is within .5 inch of 
How large does our sample size need to be?

35. Confidence Interval for 

36. Good news: we have an equation Bad news:
Need to know s
We don’t know n so we don’t know the degrees of freedom to find t*n-1

37. A Way Around this Problem: Use the Standard Normal

38. Sampling distribution of y
Confidence level
.95

39. Estimating s
Previously collected data or prior knowledge of the population
If the population is normal or near-normal, then s can be conservatively estimated by
s  range 6
99.7% of obs. Within 3  of the mean

40. We want to be 95% confident that we are within .5 inch of , so
Example: sample size to estimate mean height µ of NCSU undergrad. male students
We want to be 95% confident that we are within .5 inch of , so
MOE = .5; z*=1.96
Suppose previous data indicates that s is about 2 inches.
n= [(1.96)(2)/(.5)]2 = 61.47
We should sample 62 male students

41. Example: Sample Size to Estimate a Population Mean -Textbooks
Suppose the financial aid office wants to estimate the mean NCSU semester textbook cost  within MOE=$25 with 98% confidence. How many students should be sampled? Previous data shows  is about $85.

42. Example: Sample Size to Estimate a Population Mean -NFL footballs
The manufacturer of NFL footballs uses a machine to inflate new footballs
The mean inflation pressure is 13.0 psi, but random factors cause the final inflation pressure of individual footballs to vary from 12.8 psi to 13.2 psi
After throwing several interceptions in a game, Tom Brady complains that the balls are not properly inflated.
The manufacturer wishes to estimate the mean inflation pressure to within .025 psi with a 99% confidence interval. How many footballs should be sampled?

43. Example: Sample Size to Estimate a Population Mean 
The manufacturer wishes to estimate the mean inflation pressure to within .025 pound with a 99% confidence interval. How may footballs should be sampled?
99% confidence  z* = 2.58; ME = .025
 = ? Inflation pressures range from 12.8 to 13.2 psi
So range =13.2 – 12.8 = .4;   range/6 = .4/6 = .067 1 2 3 48

44. Required Sample Size To Estimate a Population Mean 
It is frequently the case that we are sampling without replacement.

45. Required Sample Size To Estimate a Population Mean  When Sampling Without Replacement.

46. Required Sample Size To Estimate a Population Mean  When Sampling Without Replacement.

47. Required Sample Size To Estimate a Population Mean  When Sampling Without Replacement.

48. 4.3 Estimation of population total 

49. 4.3 Estimation of population total 

50. Required Sample Size To Estimate a Population Total 

51. 4.3 Estimation of population total 
Estimate number of lakes in Minnesota, the “Land of 10,000 Lakes”.
WORKSHEET 6
Use Minnesota lakes data at statcrunch (number of lakes in each of 87 counties). MOE=500, 95% confidence. Estimate s =100.
n0 = .154; n = N^2*n0/(1+N*n0)=80.96
Calculate ybar = mean number of lakes per county, multiply by N = 87.

52. 4.5 Estimation of population proportion p
Interested in the proportion p of a population that has a characteristic of interest.
Estimate p with a sample proportion.

53. 4.5 Estimation of population proportion p

54. 4.5 Estimation of population proportion p

55. 4.5 Estimation of population proportion p

56. Required Sample Size To Estimate a Population Proportion p When Sampling Without Replacement.

57. 4.6 Comparing Estimates

58. 4.6 Comparing Estimates: Comparing Means

59. 4.6 Comparing Estimates: Comparing Means

60. Population 1 Population 2
Parameters: µ1 and  Parameters: µ2 and 22
(values are unknown) (values are unknown)
Sample size: n1 Sample size: n2
Statistics: x1 and s Statistics: x2 and s22
Estimate µ1 µ2 with x1 x2

61. An estimate of the degrees of freedom is
Sampling distribution model for ?
Shape? df
An estimate of the degrees of freedom is
min(n1 − 1, n2 − 1).

62. 4.6 Comparing Estimates: Comparing Means

63. 4.6 Comparing Estimates: Comparing Means (Special Case, Seldom Used)

64. 4.6 Comparing Estimates: Comparing Proportions, Two Cases
Difference between two polls
Differences within a single poll question
Comparing proportions for a single poll question, horse-race polls (dependent proportions)
Difference of proportions between 2 independent polls

65. 4.6 Comparing Estimates: Comparing Proportions in Two Independent Polls

66. 4.6 Comparing Estimates: Comparing Proportions in Two Independent Polls

67. Multinomial Sampling Situation
4.6 Comparing Estimates: Comparing Dependent Proportions in a Single Poll
Multinomial Sampling Situation
Typically 3 or more choices in a poll

68. Worksheet

69. End of Chapter 4