Confidence Interval Estimation Sigma Unknown — Transcript

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1. 0:01welcome to this tutorial on interval
2. 0:03estimation for the mean when Sigma is
3. 0:05unknown in the last tutorial we
4. 0:08discussed how to create an interval
5. 0:10around the mean when Sigma is known
6. 0:12however in many cases we do not have a
7. 0:15known process or historical data to
8. 0:17determine the true standard deviation
9. 0:19Sigma and therefore we have to use
10. 0:21sample data to estimate both the mean mu
11. 0:25and the standard deviation Sigma these
12. 0:27types of cases are referred to as Sigma
13. 0:30unknown
14. 0:31cases so for Sigma unknown cases the
15. 0:35topic of this tutorial we use the sample
16. 0:37standard deviation s to estimate Sigma
17. 0:41we will also use a probability
18. 0:43distribution called the T distribution
19. 0:45instead of the Z distribution this T
20. 0:48distribution accounts for samples of a
21. 0:50smaller size and when samples are used
22. 0:52to estimate population standard
23. 0:54deviations the area under a t
24. 0:57distribution looks very similar to a z
25. 0:59distribution
26. 1:00to read the area under a curve using a t
27. 1:03table we will need two values the
28. 1:06degrees of freedom and the upper tail
29. 1:09area under the curve which is Alpha
30. 1:11divided in half the degrees of freedom
31. 1:13is calculated based on the information
32. 1:16that is used to calculate s the sample
33. 1:18standard deviation for the types of
34. 1:20examples we will be doing in this
35. 1:22tutorial the degrees of freedom will be
36. 1:24given as
37. 1:26nus1 the second value you will need is
38. 1:29Alpha divided in half remember Alpha is
39. 1:33the level of significance which is 1
40. 1:36minus our level of confidence so if we
41. 1:38choose a 95% level of confidence then
42. 1:41the level of significance is 1 minus .95
43. 1:45or
44. 1:4605 and then Alpha / in half is
45. 1:51025 let's take a look at what the T
46. 1:54distribution looks
47. 1:57like here you can see three different
48. 1:59distributions two t distributions and
49. 2:02one standard normal distributions the
50. 2:05two t distributions have one with 10
51. 2:07degrees of freedom and one with 20° of
52. 2:10freedom and then you can see in purple
53. 2:12the normal distribution curve as you can
54. 2:14see the T distribution for the smaller
55. 2:17amounts of degrees of freedom which
56. 2:19usually means smaller sample sizes is
57. 2:21shorter and fatter than the normal
58. 2:24distribution as the degrees of freedom
59. 2:27increased from 10 degrees of freedom to
60. 2:2920 degrees of freedom the T distribution
61. 2:32starts to look more and more like a
62. 2:34normal distribution as the degrees of
63. 2:36freedom approaches higher and higher
64. 2:38numbers above 100 and approaching
65. 2:41Infinity then the T distribution
66. 2:43actually converges on the Z distribution
67. 2:46now let's look at what a tea table looks
68. 2:49like here is a tea
69. 2:51table at first glance it looks similar
70. 2:54to a z table but if you look more
71. 2:56closely you will see the left hand
72. 2:58column down has the degrees of freedom
73. 3:01labeled and the top row has the upper
74. 3:04tail area or Alpha divided in half using
75. 3:07those two values we can look up any area
76. 3:10under the curve we need degrees of
77. 3:12freedom and we need Alpha divided in
78. 3:15half also if you look at the last line
79. 3:17in the T table it's labeled infinity and
80. 3:20the values there correspond to the same
81. 3:23values as we found in the Z table if you
82. 3:26recall for a 95% confidence interval we
83. 3:29found a Z value of
84. 3:311.96 which is the same value you would
85. 3:33find if you look up Alpha / half or 025
86. 3:38and infinity degrees of freedom take a
87. 3:40look at Infinity degrees of freedom and
88. 3:43025 is Alpha / half and you will see
89. 3:471.96 the same value that was in the Z
90. 3:49table now if you recall from the
91. 3:52previous tutorial on interval estimation
92. 3:54for Sigma known we use this formula xar
93. 3:58plus or minus a margin of a error which
94. 4:00is z * Sigma / the < TK of n notice Z
95. 4:04now has a subscript of alpha / in half
96. 4:07the subscript can be left out but it is
97. 4:09more accurate to have it there and now
98. 4:11that we understand better what Alpha is
99. 4:14and that for confidence interval
100. 4:15estimations we need to split Alpha in
101. 4:17half then it is a good idea to have this
102. 4:20as a subscript now let's look at the
103. 4:22formula for Interval estimation for
104. 4:24Sigma unknown here is the formula it is
105. 4:27very similar to the one above except
106. 4:30note the small differences first of all
107. 4:32since we are using the formula in the
108. 4:34case of Sigma unknown we will need to
109. 4:37use S instead of Sigma and since we are
110. 4:40estimating Sigma using S then we will
111. 4:43need to look up values in the T table so
112. 4:46that we have t instead of Z in the
113. 4:48formula let's see how this works using
114. 4:50an example let's say we want to draw a
115. 4:5390% confidence interval on the true
116. 4:56average grade on a population of
117. 4:58students taking a statistics Exam assume
118. 5:01we don't know Sigma the population
119. 5:03standard deviation so we take a sample
120. 5:06of Nal 20 and we get a sample average of
121. 5:1075 and a sample standard deviation of 15
122. 5:14now since we are drawing a 90%
123. 5:16confidence interval then the level of
124. 5:18significance is 1 minus .9
125. 5:22or100 and then Alpha / in half is 05
126. 5:26degrees of freedom is nus1 so the
127. 5:29degrees of freedom noted here as DF for
128. 5:32degrees of freedom is n minus1 which is
129. 5:3520 - 1 or 19 so let's look up in the T
130. 5:39table Alpha divided half 05 and degrees
131. 5:43of freedom
132. 5:4519 and we get 1.72 n so now we are ready
133. 5:50to calculate the confidence interval
134. 5:52estimate for the mean using the formula
135. 5:55xar plus and minus t * s over theare <
136. 6:00TK of n we get
137. 6:0375 plus and minus
138. 6:061729 * s which is 15 / theare < TK of 20
139. 6:12which gives us 75 plus and Min -
140. 6:19579921 confidence interval on the mean
141. 6:22of between
142. 6:256928 and 80. 772
143. 6:30marking this off on the distribution we
144. 6:32can see visually that the lower limit is
145. 6:35around
146. 6:3669.2 and the upper limit is around 80.8
147. 6:40and this represents an interval within
148. 6:42which we are 90% confident that the true
149. 6:44mean exists remember we don't know what
150. 6:48the true mean is this represents an
151. 6:50interval within which we can be a
152. 6:52certain percent confident in this case
153. 6:5490% confident that the true mean is
154. 6:57located somewhere within this interval
155. 6:59the tail areas represent the error that
156. 7:02is there is a 10% chance that the true
157. 7:04mean is outside this interval and half
158. 7:07of that Arrow will be in the upper tail
159. 7:08region above 80.8 and half of that Arrow
160. 7:12will be in the lower tail area below
161. 7:1669.2 that concludes this tutorial on
162. 7:19drawing confidence intervals when Sigma
163. 7:22is unknown I hope you enjoyed this
164. 7:24tutorial and I hope you learned
165. 7:28something