1. Welcome to .
Week 08 Thurs .
MAT135 Statistics

2. We use the sample mean 𝒙 to estimate the unknown population mean µ
Estimation
We use the sample mean 𝒙 to estimate the unknown population mean µ

3. Estimation
Using the sample mean 𝒙 to estimate the unknown population mean µ is called “making inferences”

4. Estimation
The sample standard deviation “s” is the best estimate we have for the unknown population standard deviation “σ”

5. Using s to estimate σ is also an inference
Estimation
Using s to estimate σ is also an inference

6. s IS NOT the measure of variability in the new population of 𝒙 s
Estimation
s IS NOT the measure of variability in the new population of 𝒙 s

7. It needs to be decreased to take sample size into account!
Estimation
It needs to be decreased to take sample size into account!

8. Estimation
We use: s/ n for the measure of variability in the new population of 𝒙 s

9. Estimation
The standard deviation of the 𝒙 s: s/ n is called the “standard error” abbreviated “se”

10. Estimation
So our curve is:
𝒙 -3se 𝒙 -2se 𝒙 -se 𝒙 𝒙 +se 𝒙 +2se 𝒙 +3se

11. Binomial Proportions
All of these activities can be done when you have measured data (continuous)

12. Binomial Proportions
What if you had counts in two categories and wanted to estimate the population proportion?

13. Binomial Proportions Weep? Throw things? Give up?

14. Binomial Proportions Naw… we’ve got an easy solution!

15. Binomial Proportions
Suppose you have a random sample of size “n”

16. Binomial Proportions
Suppose you have a random sample of size “n” Suppose you want the proportion of successes p

17. Binomial Proportions
Suppose you have a random sample of size “n” Suppose you know the number of successes in your sample is “x”

18. Binomial Proportions
The sample proportion (symbolized: p , called: “p-hat”) is: p = x n

19. You have a random sample with frequencies: Is it binomial?
BINOMIAL PROPORTIONS
IN-CLASS PROBLEM 1
You have a random sample with frequencies: Is it binomial?
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20. You have a random sample with frequencies: Is it binomial? yep
BINOMIAL PROPORTIONS
IN-CLASS PROBLEM 1
You have a random sample with frequencies: Is it binomial? yep
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21. You have a random sample with frequencies: What is n?
BINOMIAL PROPORTIONS
IN-CLASS PROBLEM 2
You have a random sample with frequencies: What is n?
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22. You have a random sample with frequencies: What is n? 58
BINOMIAL PROPORTIONS
IN-CLASS PROBLEM 2
You have a random sample with frequencies: What is n? 58
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23. You have a random sample with frequencies: What is xred?
BINOMIAL PROPORTIONS
IN-CLASS PROBLEM 3
You have a random sample with frequencies: What is xred?
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24. You have a random sample with frequencies: What is xred? 23
BINOMIAL PROPORTIONS
IN-CLASS PROBLEM 3
You have a random sample with frequencies: What is xred? 23
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25. You have a random sample with frequencies: What is p red?
BINOMIAL PROPORTIONS
IN-CLASS PROBLEM 4
You have a random sample with frequencies: What is p red?
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26. BINOMIAL PROPORTIONS
IN-CLASS PROBLEM 4
You have a random sample with frequencies: What is p red? x n = ≈ 40.0%
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27. Questions?

28. Binomial Estimation
Just like the sample mean 𝒙 is the best estimate of the true population mean μ…

29. Binomial Estimation
…there is a true population proportion “p” that is best estimated by the sample proportion p

30. Binomial Estimation
In symbols: µ p = p

31. BINOMIAL ESTIMATION
IN-CLASS PROBLEM 5
You have a random sample with frequencies: What is the best estimate for pred?
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32. BINOMIAL ESTIMATION
IN-CLASS PROBLEM 5
You have a random sample with frequencies: What is the best estimate for pred? 40.0%
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33. BINOMIAL ESTIMATION
IN-CLASS PROBLEM 6
You have a random sample with frequencies: What is the best estimate for qred?
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34. BINOMIAL ESTIMATION
IN-CLASS PROBLEM 6
You have a random sample with frequencies: What is the best estimate for qred? 60.0%
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35. Binomial Estimation
Because p-values distributions are often skewed, you need to be sure your sample size is large before assigning probabilities to this estimate using a normal distribution

36. Binomial Estimation
You can assume normality if: np ≥ 5or10 and n(1-p) ≥ 5or10 (nq ≥ 5or10)

37. This is known as the Rule of Sample Proportions
Binomial Estimation
This is known as the Rule of Sample Proportions

38. Binomial Estimation
Note: your book says the shape of the sampling distribution of p-hat is approximately normal provided np(1-p)≥10 (I think it’s a typo)

39. Binomial Sampling
What if your sample size is too small?

40. Binomial Sampling
The normal approximation can always be used, but if the conditions are not met, then the approximation may not be that good of an approximation.

41. BINOMIAL ESTIMATION
IN-CLASS PROBLEM 7
You have a random sample with frequencies: Is p red likely to be normally-distributed?
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42. BINOMIAL ESTIMATION
IN-CLASS PROBLEM 7
You have a random sample with frequencies: Is p red likely to be normally-distributed? Is np ≥ 5or10?
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43. BINOMIAL ESTIMATION
IN-CLASS PROBLEM 7
You have a random sample with frequencies: Is p red likely to be normally-distributed? Is np ≥ 5or10? np ≈ 58 × .400 ≈ 23.2
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44. BINOMIAL ESTIMATION
IN-CLASS PROBLEM 7
You have a random sample with frequencies: Is p red likely to be normally-distributed? Is nq ≥ 5or10?
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45. BINOMIAL ESTIMATION
IN-CLASS PROBLEM 7
You have a random sample with frequencies: Is p red likely to be normally-distributed? Is nq ≥ 5or10? nq ≈ 58 ×.400 ≈ 23.2
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46. BINOMIAL ESTIMATION
IN-CLASS PROBLEM 7
You have a random sample with frequencies: Is p red likely to be normally-distributed?
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47. BINOMIAL ESTIMATION
IN-CLASS PROBLEM 7
You have a random sample with frequencies: Is p red likely to be normally-distributed? yep, close enough
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48. Questions?

49. Binomial Sampling
Just as a gazillion samples each with its own mean gives a new population: the gazillion means

50. Binomial Sampling
…a gazillion samples each with its own proportion gives a new population: the gazillion p s

51. Binomial Sampling
And, again if you plotted the frequency of the gazillion p values, it would be called a SAMPLING DISTRIBUTION

52. Binomial Sampling
And, again, the shape of the plot of the gazillion sample means would have a normal-ish distribution NO MATTER WHAT THE ORIGINAL DATA LOOKED LIKE

53. Binomial Sampling
Very non- normal population

54. Binomial Sampling
Very non- normal population Normal-er

55. Binomial Sampling
Very non- normal population Normal-ish

56. Binomial Sampling
And, again, as “n” increases, the variability (spread) decreases

57. Binomial Sampling
The standard deviation of the p s is: σ p = p(1−p) n = pq n

58. Binomial Sampling
The standard deviation of the p s is: σ p = p(1−p) n = pq n also called the standard error of p-hat: se p

59. Binomial Sampling So a normal curve would be:
p -3 pq n p -2 pq n p - pq n p p + pq n p +2 pq n p +3 pq n

60. Binomial Sampling So a normal curve would be:
p -3 se p p -2 se p p -se p p p +se p p +2 se p p +3 se p

61. BINOMIAL SAMPLING
IN-CLASS PROBLEM 8
You have a random sample with frequencies: What is the standard error of p-hat?
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62. BINOMIAL SAMPLING
IN-CLASS PROBLEM 8
You have a random sample with frequencies: What is the standard error of p-hat? × se p = pq n = .6×.4 58 ≈ .064
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63. BINOMIAL SAMPLING
IN-CLASS PROBLEM 9
You have a random sample with frequencies: What would the normal curve look like?
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64. BINOMIAL SAMPLING IN-CLASS PROBLEM 9
p -3 se p p -2 se p p -se p p p +se p p +2 se p p +3 se p

65. BINOMIAL SAMPLING IN-CLASS PROBLEM 9
.6-3(.064) .6-2(.064) (.064) .6+3(.064)

66. BINOMIAL SAMPLING IN-CLASS PROBLEM 9

67. What is P(0.536<p<.664)? BINOMIAL SAMPLING IN-CLASS PROBLEM 10

68. What is P(0.536<p<.664)? 68% BINOMIAL SAMPLING
IN-CLASS PROBLEM 10
What is P(0.536<p<.664)? 68%

69. BINOMIAL SAMPLING
IN-CLASS PROBLEM 11
What is the range of values for p that would with 95% certainty include the true p?

70. BINOMIAL SAMPLING
IN-CLASS PROBLEM 11
What is the range of values for p that would with 95% certainty include the true p? .472<p<.728

71. What is the probability the true p lies between 0.47 and .65?
BINOMIAL SAMPLING
IN-CLASS PROBLEM 12
What is the probability the true p lies between 0.47 and .65?

72. What is the probability the true p lies between 0.47 and .65? 76.2%
BINOMIAL SAMPLING
IN-CLASS PROBLEM 12
What is the probability the true p lies between 0.47 and .65? 76.2%

73. Binomial Sampling
A study I did for my science class…

74. Binomial Sampling
Some diamonds fluoresce under UV light

75. Binomial Sampling
What percentage of diamonds fluoresce?

76. Binomial Sampling
I inherited a bunch of jewelry Some of it has diamonds in it

77. Binomial Sampling
Is this a binomial variable?

78. Binomial Sampling
What is n?
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79. Binomial Sampling
What is x?
Fluoresce
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80. Binomial Sampling What is your best estimate of p? Fluoresce Don’t 8
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81. Binomial Sampling
Is it normal?
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82. Binomial Sampling
To find out what sample size you need to: Estimate p Use the min of p and q Algebra magic: n > 5or10/min(p,q)

83. Binomial Sampling
p ≈ .066 so q ≈ .934 so we’ll use p n > 10/.066 or 151.5
Fluoresce
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84. Binomial Sampling
p ≈ .066 so q ≈ .934 so we’ll use p n > 10/.066 or In other words n > 152
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85. Questions?

86. You survived! Turn in your homework! Don’t forget your homework
due next week!
Have a great
rest of the week!