Sample size for a given margin of error for a mean | AP Statistics | Khan Academy — Transcript

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1. 0:00nadia wants to create a confidence
2. 0:02interval to estimate the mean driving
3. 0:05range for her company's new electric
4. 0:07vehicle
5. 0:08she wants the margin of error to be no
6. 0:11more than 10 kilometers at a 90 percent
7. 0:14level of confidence a pilot study
8. 0:17suggests that the driving ranges for
9. 0:19this type of vehicle have a standard
10. 0:21deviation of 15 kilometers
11. 0:24which of these is the smallest
12. 0:26approximate sample size required to
13. 0:29obtain
14. 0:30the desired margin of error so pause
15. 0:33this video and see if you can think
16. 0:35about this on your own
17. 0:38so the traditional way that we would
18. 0:39construct a margin of error on a
19. 0:42confidence interval we take a sample and
20. 0:44from that sample we construct
21. 0:47the mean
22. 0:48and then we add or subtract a margin of
23. 0:51error around that to construct the
24. 0:54confidence interval and the way that
25. 0:55we've done that since we're dealing with
26. 0:57means is we say all right if we don't
27. 1:00know the standard deviation of the
28. 1:01population it's appropriate to use the t
29. 1:05statistic so our critical value we
30. 1:07denote as t star and you'd multiply that
31. 1:10times the sample standard deviation
32. 1:13divided by the square root of
33. 1:17your sample size
34. 1:19now this question is all about what is
35. 1:22an appropriate sample size given that we
36. 1:25want to have a 90
37. 1:26level of confidence and what's tricky
38. 1:29here is when you're using a t-table
39. 1:31right over here not only do you need to
40. 1:33know the 90
41. 1:34level of confidence you also need to
42. 1:36know the degrees of freedom and the
43. 1:38degrees of freedom is going to be n
44. 1:40minus 1 but we don't know what that's
45. 1:41going to be without knowing n so how
46. 1:43would we determine an n
47. 1:45similarly you don't know what your
48. 1:46sample standard deviation is going to be
49. 1:48until you actually take some samples so
50. 1:51instead of that what we could think
51. 1:52about we know that another legitimate
52. 1:56way to construct a confidence interval
53. 1:57and the margin of error is to say all
54. 2:00right i can take my sample mean and i
55. 2:03can add or subtract
56. 2:05a
57. 2:06z-score a critical value and this time
58. 2:08use the z table where if i multiply that
59. 2:10times the true
60. 2:12population standard deviation and divide
61. 2:15that by the square root of n
62. 2:18now you might say well i don't know the
63. 2:19true population standard deviation but
64. 2:21they tell us a pilot study suggests that
65. 2:24the driving ranges for this type of
66. 2:26vehicle have a standard deviation of 15
67. 2:28kilometers so we could use this as an
68. 2:30estimate of our true population standard
69. 2:32deviation so this is 15 kilometers right
70. 2:36over here and then the good thing about
71. 2:38a z table is you don't have to think
72. 2:40about degrees of freedom you could just
73. 2:42look up your confidence interval and so
74. 2:44then we could just say
75. 2:46that look z star
76. 2:49times 15 kilometers
77. 2:51over our square root of n
78. 2:53this right over here is our margin of
79. 2:55error
80. 2:56this right over here is our margin of
81. 2:57error that has to be no more than 10
82. 2:59kilometers
83. 3:01so that has to be a less than or equal
84. 3:03to
85. 3:0410.
86. 3:05and we can figure out what z star needs
87. 3:07to be for a 90 percent confidence level
88. 3:09and so then we just solve for n so let's
89. 3:11do that
90. 3:12now to figure out z star i could use a z
91. 3:14table but just for diversity of
92. 3:16methodology let's use a a calculator
93. 3:19here
94. 3:20so to figure out the z value that would
95. 3:22give us a 90 confidence interval i can
96. 3:24use a function called
97. 3:26inverse norm and you can see that right
98. 3:28over here that's choice 3. let me just
99. 3:30select that and what it'll do is
100. 3:34you give it the area that you want under
101. 3:35a normal curve you can even specify the
102. 3:38mean and the standard deviation although
103. 3:39you want the mean to be 0 and the
104. 3:41standard deviation to be 1 if you really
105. 3:43want to figure out a z-score here and so
106. 3:46it'll do is it will give you the z-score
107. 3:49that will give you that corresponding
108. 3:50area
109. 3:51and so i want and actually it's already
110. 3:53selected that i want the center
111. 3:55area to be 90
112. 3:57so i could say point
113. 4:00right over here if i use the left tail
114. 4:03then that means if i have 90 percent of
115. 4:05the center that means i would have five
116. 4:06percent of either tail so instead of
117. 4:07doing it .9 and center i could have done
118. 4:10.05 and used the left tail or used .05
119. 4:14and use the right tail but this is
120. 4:16exactly what i want so let me just go
121. 4:18and paste this
122. 4:20and so this should give me the
123. 4:23appropriate
124. 4:24z value
125. 4:27so there you go if i want this middle
126. 4:29ninety percent the center ninety percent
127. 4:31i have to go one point roughly one point
128. 4:34six four
129. 4:36five standard deviations below the mean
130. 4:37and that same amount above the mean so
131. 4:39it's roughly our critical value here is
132. 4:42approximately 1.6
133. 4:44let's just say
134. 4:451.645 so we have
135. 4:491.645
136. 4:52times
137. 4:54times 15 over the square root of n is
138. 4:57going to be less than or equal to 10.
139. 5:00and so now there's a couple of ways that
140. 5:02you could do this we could do a little
141. 5:03bit of algebra to simplify this
142. 5:05inequality i encourage you to do so or
143. 5:07you could even try out some values here
144. 5:09and see which of these ends would make
145. 5:11this true and we want the smallest
146. 5:13possible one i'll do it the algebraic
147. 5:15way because if you're actually doing
148. 5:17this in the real world nadia would not
149. 5:19have a multiple choice right over here
150. 5:20she'd have to figure out the sample size
151. 5:22in order to conduct her study
152. 5:24so let's do that
153. 5:26so let's see if i divide both sides by
154. 5:291.645 and 15 what do i get i get 1 over
155. 5:34the square root of n
156. 5:36is less than or equal to
157. 5:3910
158. 5:41over
159. 5:421.645
160. 5:46over
161. 5:47over
162. 5:4815.
163. 5:50and then if i take the reciprocal of
164. 5:51both sides i get the square root of n
165. 5:54is greater than or equal to from taking
166. 5:56the reciprocal of both sides and so this
167. 5:58is going to be
168. 6:021.645
169. 6:04times 15 times
170. 6:0815 all of that over 10
171. 6:10all of that over 10. see 15 over 10 is
172. 6:13just 1.5 so let me just write that as
173. 6:151.5 right over here and then if i square
174. 6:18both sides i would get that n needs to
175. 6:22be greater than or equal to
176. 6:261.645
177. 6:28times
178. 6:30times 1.5 and then all of that squared i
179. 6:34just squared both sides all of that
180. 6:36squared and so let's get our calculator
181. 6:37back we are going to have 1.645
182. 6:43times
183. 6:441.5 and then we want to square it and we
184. 6:48get 6 point approximately 6.0
185. 6:526.09
186. 6:56so n
187. 6:57has to be greater than or equal to
188. 6:596.09 and of course our sample size needs
189. 7:02to actually be a whole number so what's
190. 7:04the smallest whole number that is larger
191. 7:06than 6.09 well that's going to be 7. so
192. 7:10that would be this choice right over
193. 7:12here this is the smallest approximate
194. 7:14sample size required to obtain the
195. 7:16desired margin of error and of course we
196. 7:19won't really really know until we
197. 7:21actually conduct the study we obviously
198. 7:23here use an estimate of the population
199. 7:26standard deviation we used and we used a
200. 7:30z table but it will be interesting when
201. 7:33nadia actually conducts the study to see
202. 7:34if her margin of error is indeed no more
203. 7:37than 10 kilometers with a 90 level of
204. 7:40confidence