1. Welcome to .
Week 10 Thurs .
MAT135 Statistics

2. In-Class Project
Is normal human temperature REALLY 98.6° ?

3. Normal Distribution
The most popular continuous graph in statistics is the NORMAL DISTRIBUTION

4. Normal Distribution
We use normal distributions a lot in statistics because lots of things have graphs this shape!
heights
weights
IQ test scores
bull’s eyes

5. Sampling Distributions
Even data which are not normally distributed have sample averages which DO have normal distributions

6. Sampling Distributions
As “n” increases, the distributions of the sample means become closer and closer to normal

7. Sampling Distributions
We usually say the sample mean will be normally distributed if n is ≥ 20or30 (the “good-enuff” value)

8. Sampling Distributions
We will use the sample mean 𝒙 to estimate the unknown population mean µ

9. Inferences about μ
Using the sample mean 𝒙 to estimate the unknown population mean µ is called “making inferences”

10. Inferences about μ
If you can assume the distribution of the sample means is normal, you can use the normal distribution probabilities for making probability statements about µ

11. as “n” increases, variability (spread) also decreases
Inferences about μ
as “n” increases, variability (spread) also decreases

12. Inferences about μ
We use: s/ n for the measure of variability in the new population of 𝒙 s

13. Inferences about μ
The standard deviation of the 𝒙 s: s/ n is called the “standard error” abbreviated “se”

14. Inferences about μ 𝒙 -3se 𝒙 -2se 𝒙 -se 𝒙 𝒙 +se 𝒙 +2se 𝒙 +3se
So our normal curve for the true value of the population mean µ is:
𝒙 -3se 𝒙 -2se 𝒙 -se 𝒙 𝒙 +se 𝒙 +2se 𝒙 +3se

15. Inferences about μ
About 95% of the possible values for μ will be within 2 SE of 𝒙

16. This allows us to create a “confidence interval” for values of μ
Confidence Intervals
This allows us to create a “confidence interval” for values of μ

17. Confidence Intervals
Confidence interval formula: 𝒙 - 2s/ n ≤ μ ≤ 𝒙 + 2s/ n or 𝒙 - 2se ≤ μ ≤ 𝒙 + 2se With a confidence level of 95%

18. The “2” in the equations is called the “critical value”
Confidence Intervals
The “2” in the equations is called the “critical value”

19. It comes from the normal curve, which gives us the 95%
Confidence Intervals
It comes from the normal curve, which gives us the 95%

20. 2s/ n or 2se is called the “margin of error”
Confidence Intervals
2s/ n or 2se is called the “margin of error”

21. What if we wanted a confidence level of 99%
Confidence Intervals
PROJECT QUESTION
What if we wanted a confidence level of 99%

22. Confidence Intervals
PROJECT QUESTION
What if we wanted a confidence level of 99% We’d use a value of “3” rather than 2

23. Confidence Intervals
For most scientific purposes, 95% is “good-enuff” In the law, 98% is required for a criminal case In medicine, 99% is required

24. Confidence Intervals
For a 95% confidence interval, 95% of the values of μ will be within 2se of 𝒙

25. Confidence Intervals
If we use the confidence interval to estimate a likely range for true values of μ, we will be right 95% of the time

26. For a 95% confidence interval, we will be WRONG 5% of the time
Confidence Intervals
For a 95% confidence interval, we will be WRONG 5% of the time

27. For a 99% confidence interval, how much of the time will we be wrong?
Confidence Intervals
PROJECT QUESTION
For a 99% confidence interval, how much of the time will we be wrong?

28. Confidence Intervals
PROJECT QUESTION
For a 99% confidence interval, how much of the time will we be wrong? we will be wrong 1% of the time

29. Confidence Intervals
The percent of time we are willing to be wrong is called “α” (“alpha”) or “the α-level”

30. Confidence Intervals
Everyday use of confidence intervals: You will frequently hear that a poll has a candidate ahead by 10 points with a margin of error of 3 points

31. Confidence Intervals
This means: 10-3 ≤ true difference ≤ 10+3 Or, the true difference is between 7 and 13 points (with 95% likelihood)

32. Confidence Intervals
PROJECT QUESTION
Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 Can we assume normality?

33. Confidence Intervals
PROJECT QUESTION
Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 Can we assume normality? yes, because n>20

34. Confidence Intervals
PROJECT QUESTION
Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the α-level?

35. Confidence Intervals
PROJECT QUESTION
Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the α-level? 5%

36. Confidence Intervals
PROJECT QUESTION
Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the critical value?

37. Confidence Intervals
PROJECT QUESTION
Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the critical value? 2, because we want a 95% confidence interval

38. Confidence Intervals
PROJECT QUESTION
Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the standard error?

39. Confidence Intervals
PROJECT QUESTION
Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the standard error? s/ n = 5/ 25 = 5/5 = 1

40. Confidence Intervals
PROJECT QUESTION
Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the margin of error?

41. Confidence Intervals
PROJECT QUESTION
Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the margin of error? 2se = 2(1) = 2

42. Confidence Intervals
PROJECT QUESTION
Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the confidence interval?

43. Confidence Intervals
PROJECT QUESTION
Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the confidence interval? 𝒙 - 2s/ n ≤ μ ≤ 𝒙 + 2s/ n 7 – 2 ≤ μ ≤ ≤ μ ≤ 9 with 95% confidence

44. Confidence Intervals
PROJECT QUESTION
Interpreting confidence intervals: If the 95% confidence interval is: 5 ≤ µ ≤ 9 Is it likely that µ = 10?

45. Confidence Intervals
PROJECT QUESTION
No, because it’s outside of the interval That would only happen 5% of the time

46. Find the 95% confidence interval for μ given: 𝒙 = 53 s = 14 n = 49
Confidence Intervals
PROJECT QUESTION
Find the 95% confidence interval for μ given: 𝒙 = 53 s = 14 n = 49

47. Confidence Intervals
PROJECT QUESTION
Find the 95% confidence interval for μ given: 𝒙 = 53 s = 14 n = ≤ µ ≤ 57

48. Can you say with 95% confidence that µ ≠ 55?
Confidence Intervals
PROJECT QUESTION
Can you say with 95% confidence that µ ≠ 55?

49. Confidence Intervals
PROJECT QUESTION
Can you say with 95% confidence that µ ≠ 55? Nope… it’s in the interval It IS a likely value for µ

50. Find the 95% confidence interval for μ given: 𝒙 = 481 s = 154 n = 121
Confidence Intervals
PROJECT QUESTION
Find the 95% confidence interval for μ given: 𝒙 = 481 s = 154 n = 121

51. Confidence Intervals
PROJECT QUESTION
Find the 95% confidence interval for μ given: 𝒙 = 481 s = 154 n = ≤ µ ≤ 509

52. Can you say with 95% confidence that µ might be 450?
Confidence Intervals
PROJECT QUESTION
Can you say with 95% confidence that µ might be 450?

53. Confidence Intervals
PROJECT QUESTION
Can you say with 95% confidence that µ might be 450?
µ is unlikely to be 450 – that value is outside of the confidence interval and would only happen 5% of the time

54. You will have a smaller interval if you have a larger value for n
Confidence Intervals
You will have a smaller interval if you have a larger value for n

55. So you want to take the LARGEST sample you can
Confidence Intervals
So you want to take the LARGEST sample you can

56. This is called the “LAW OF LARGE NUMBERS”
Confidence Intervals
This is called the “LAW OF LARGE NUMBERS”

57. Confidence Intervals
LAW OF LARGE NUMBERS The larger your sample size, the better your estimate

58. What if you have a sample size smaller than 20???
Confidence Intervals
What if you have a sample size smaller than 20???

59. Confidence Intervals What if you have a sample size smaller than 20???
You must use a different (bigger) critical value
W.S. Gosset 1908

60. Questions?

61. “Tailed” Tests
A two-tailed test will reject H0 either if the experimental values we get are too high or too low

62. “Tailed” Tests
α is split between the upper and lower tails

63. “Tailed” Tests
A one-tailed test will reject H0 only on the side we think is likely to be true

64. “Tailed” Tests
You will be able to reject H0 more often for a one-tailed test – if you pick the right tail!

65. “TAILED” TESTS
PROJECT QUESTION
Your owner's manual says you should be getting 30 mpg highway After owning the car for six months, you are only getting 27 mpg highway

66. “TAILED” TESTS
PROJECT QUESTION
Is that different enough to reject the company's claim? What is your α-level? What is Ha? What is H0?

67. “TAILED” TESTS
PROJECT QUESTION
Is that different enough to reject the company's claim? What is your α-level? 5% or 0.05 What is Ha? What is H0?

68. “TAILED” TESTS
PROJECT QUESTION
Is that different enough to reject the company's claim? What is your α-level? 5% or 0.05 What is Ha? μ < 30 mpg What is H0?

69. “TAILED” TESTS
PROJECT QUESTION
Is that different enough to reject the company's claim? What is your α-level? 5% or 0.05 What is Ha? μ < 30 mpg What is H0? μ ≥ 30 mpg

70. We could also write it as: H0: μ ≥ 30 mpg Ha: μ < 30 mpg
“TAILED” TESTS
PROJECT QUESTION
We could also write it as: H0: μ ≥ 30 mpg Ha: μ < 30 mpg

71. Is this a one-tailed or a two- tailed test?
“TAILED” TESTS
PROJECT QUESTION
Is this a one-tailed or a two- tailed test?

72. “TAILED” TESTS
PROJECT QUESTION
Is this a one-tailed or a two- tailed test? one-tailed Is it right-tailed or left-tailed?

73. Is it right-tailed or left-tailed? left-tailed
“TAILED” TESTS
PROJECT QUESTION
Is it right-tailed or left-tailed? left-tailed

74. “Tailed” Tests
Remember our two type of errors: Type 1 error: reject a true H0 (α) Type 2 error: fail to reject a false H0 (β)

75. “Tailed” Tests
The likelihood of making the right decision and rejecting the (false) null hypothesis is: 1 - β

76. “Tailed” Tests
The likelihood of making the right decision and rejecting the (false) null hypothesis is: 1 - β called the “power of the test”

77. “Tailed” Tests
For a given α value, we would like the test to be as "powerful" as possible, give us the best chance of rejecting a false null hypothesis

78. Which is more powerful, a one-tailed or a two-tailed test?
“TAILED” TESTS
PROJECT QUESTION
Which is more powerful, a one-tailed or a two-tailed test?

79. “TAILED” TESTS
PROJECT QUESTION
Which is more powerful, a one-tailed or a two-tailed test? one-tailed (if you guess the correct side)

80. Questions?

81. Hypothesis Tests
Now we will create a confidence interval about 𝒙 and see if our hypothesized value for μ falls in it

82. Hypothesis Tests
How to do it!

83. Hypothesis Tests
How to do it! Set your α-level (how often you are willing to be wrong)

84. Hypothesis Tests
How to do it! Set your α-level Define your Ha and H0

85. Hypothesis Tests
How to do it! Set your α-level Define your Ha and H0 Get your data (for a confidence interval, you need the hypothesized μ, s and n (or se)

86. Hypothesis Tests
How to do it! Set your α-level Define your Ha and H0 Get your data Find your critical value (for α=5% two-sided it is ≈2 for one-sided it is ≈1.64)

87. Hypothesis Tests
How to do it! Set your α-level Define your Ha and H0 Get your data Find your critical value Calculate the confidence interval for μ

88. Hypothesis Tests
How to do it! Set your α-level Define your Ha and H0 Get your data Find your critical value Calculate the confidence interval for μ The test will be: Is x in it?

89. Hypothesis Tests
PROJECT QUESTION
Back to our mpg! H0: μ ≥ 30 mpg Ha: μ < 30 mpg x = 27 And suppose we know that: se = 4 mpg

90. Hypothesis Tests
PROJECT QUESTION
H0: μ ≥ 30 mpg Ha: μ < 30 mpg x = 27 se = 4 mpg Are we going to reject H0 for values of x greater than 30 or less than 30?

91. Hypothesis Tests
PROJECT QUESTION
H0: μ ≥ 30 mpg Ha: μ < 30 mpg x = 27 se = 4 mpg The 1-tailed critical value is: 1.64

92. Hypothesis Tests
PROJECT QUESTION
H0: μ ≥ 30 mpg Ha: μ < 30 mpg x = 27 se = 4 mpg What is the margin of error?

93. Hypothesis Tests
PROJECT QUESTION
H0: μ ≥ 30 mpg Ha: μ < 30 mpg x = 27 se = 4 mpg What is the margin of error? 1.64×4 = 6.56

94. H0: μ ≥ 30 mpg Ha: μ < 30 mpg x = 27 se = 4 mpg What is the CI?
Hypothesis Tests
PROJECT QUESTION
H0: μ ≥ 30 mpg Ha: μ < 30 mpg x = 27 se = 4 mpg What is the CI?

95. Hypothesis Tests
PROJECT QUESTION
H0: μ ≥ 30 mpg Ha: μ < 30 mpg x = 27 se = 4 mpg What is the CI? 95% of x values from a population with μ ≥ 30 will fall above: 30 – 6.56 = 23.44

96. Hypothesis Tests
PROJECT QUESTION
H0: μ ≥ 30 mpg Ha: μ < 30 mpg x = 27 se = 4 mpg So reject H0 if x < What is our conclusion?

97. Hypothesis Tests
PROJECT QUESTION
H0: μ ≥ 30 mpg Ha: μ < 30 mpg x = 27 se = 4 mpg So reject H0 if x < What is our conclusion? fail to reject H0

98. Hypothesis Tests
If you reject H0 with an α-level of 0.05, we also say our x value is “significant at the .05 level” or we say we found a “significant difference”

99. Questions?

100. CI for Proportions We’ll use the normal curve for proportions:
p -3 pq n p -2 pq n p - pq n p p + pq n p +2 pq n p +3 pq n

101. If p = .4 and n = 30 find the 95% CI for p
CI FOR PROPORTIONS
PROJECT QUESTION
If p = .4 and n = 30 find the 95% CI for p

102. If p = .4 and n = 30 find the 95% CI for p se =
CI FOR PROPORTIONS
PROJECT QUESTION
If p = .4 and n = 30 find the 95% CI for p se =

103. If p = .4 and n = 30 find the 95% CI for p se = pq n = .4x.6 30 ≈ .089
CI FOR PROPORTIONS
PROJECT QUESTION
If p = .4 and n = 30 find the 95% CI for p se = pq n = .4x.6 30 ≈ .089

104. If p = .4 and n = 30 find the 95% CI for p me =
CI FOR PROPORTIONS
PROJECT QUESTION
If p = .4 and n = 30 find the 95% CI for p me =

105. If p = .4 and n = 30 find the 95% CI for p me = 2 × .089 = .178
CI FOR PROPORTIONS
PROJECT QUESTION
If p = .4 and n = 30 find the 95% CI for p me = 2 × .089 = .178

106. If p = .4 and n = 30 find the 95% CI for p CI:
CI FOR PROPORTIONS
PROJECT QUESTION
If p = .4 and n = 30 find the 95% CI for p CI:

107. CI FOR PROPORTIONS
PROJECT QUESTION
If p = .4 and n = 30 find the 95% CI for p CI: ≤ p ≤ ≤ p ≤ .578

108. Questions?

109. Hypothesis Tests
We can make our x more likely to be significant by (as usual): TAKING A LARGER SAMPLE SIZE

110. Hypothesis Tests
Because we can “cheat the system” by taking a huge sample size that will find any teeny, tiny difference to be significant, we have a backup plan

111. Hypothesis Tests
We also set levels of “practical significance” - what numerical difference would convincingly show a significant difference

112. Hypothesis Tests
These levels of practical significance come from our knowledge of the variables we are measuring

113. Hypothesis Tests
If we had taken a sample of 10,000,000 to calculate our mpg average and se, we could easily have had an se of 0.1 mpg Probably we wouldn’t really think that was a significant difference in mileage

114. Hypothesis Tests
A practically significant difference would be the amount in mpg that you would think is different enough from 30 mpg to be important

115. Hypothesis Tests
We set a level of practical significance at the same time we set the α-level

116. Hypothesis Tests
PROJECT QUESTION
What would be your level of practically significant difference for mpg?

117. Questions?

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