Confidence Intervals for a Proportion: Determining the Minimum Sample Size — Transcript

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1. 0:02let's discuss determining the minimum
2. 0:03sample size when estimating a
3. 0:07proportion suppose we are about to draw
4. 0:09a sample and we wish to estimate the
5. 0:11population proportion P we may wish to
6. 0:14estimate P to within an amount M with
7. 0:1695% confidence we may wish to estimate P
8. 0:20to within 005 or 02 or within
9. 0:24.001 or whatever we feel is appropriate
10. 0:27in a given situation
11. 0:30and the question here is how large of a
12. 0:32sample size is required to achieve this
13. 0:35in almost all situations sampling
14. 0:37doesn't come without some sort of cost
15. 0:40in terms of time money or other
16. 0:42resources so we may want to determine
17. 0:44the minimum sample size required to
18. 0:46achieve our
19. 0:50goals wanting to estimate the population
20. 0:52proportion P to within M at a certain
21. 0:55level of confidence is the same as
22. 0:56wanting the margin of error to be no
23. 0:58more than m
24. 1:00here's the margin of error of a 95%
25. 1:03confidence interval for the population
26. 1:04proportion p and we'd like this to be no
27. 1:07more than m in the general case in
28. 1:10situations like this we simply solve for
29. 1:13the sample size n but we run into a bit
30. 1:16of a problem here as the sample
31. 1:18proportion P hat appears in this formula
32. 1:20and we haven't drawn the sample yet so
33. 1:22we don't know the value of P hat and we
34. 1:25can't use it in the
35. 1:27formula so here I'm going to replace P
36. 1:30hat with P star for the time being P
37. 1:34star represents an estimated value of
38. 1:36the population proportion P we'll
39. 1:38discuss what value to use for p star in
40. 1:41a moment or
41. 1:43two now we need to isolate the sample
42. 1:45size n I'll let you work through the
43. 1:48details of the algebra but we'd isolate
44. 1:50the sample size n by multiplying both
45. 1:53sides by the square root of n dividing
46. 1:56both sides by the margin of error M and
47. 1:59then squaring both
48. 2:04sides we end up with this to estimate P
49. 2:07within M with 95% confidence we would
50. 2:10need a sample size of at least this many
51. 2:13observations to change the confidence
52. 2:16level from 95% to something else we
53. 2:19change the Zed
54. 2:23value in the general case to estimate P
55. 2:26to within M with 1 - Alpha * 100%
56. 2:30confidence we need a sample size of at
57. 2:32least this
58. 2:33quantity Z sub Alpha over 2 is the usual
59. 2:37Zed value for the given confidence level
60. 2:401.96 for a 95% confidence level 1.645
61. 2:45for a 90% confidence level
62. 2:51Etc what value should we use for
63. 2:54pstar there are two main options we may
64. 2:57have an estimate of P based on prior
65. 2:59information
66. 3:00we may have other samples from the
67. 3:02population or some past experience that
68. 3:04gives us a reasonable ballpark estimate
69. 3:06of
70. 3:07P but if we don't have a reasonable
71. 3:09estimate of P we should use a p star
72. 3:11value of
73. 3:120.5 this is a conservative or worst case
74. 3:16approach and ensures that the sample
75. 3:17size is large enough regardless of what
76. 3:19the real value of p is the quantity P
77. 3:23startimes 1 minus P star is greatest
78. 3:25when P star is equal
79. 3:27to.5 so the choice of P star of .5
80. 3:31results in the largest minimum sample
81. 3:33size of any value of P star I'll let you
82. 3:36verify that for
83. 3:38yourself let's work through a couple of
84. 3:43examples suppose we wish to estimate the
85. 3:45proportion of adults in Ontario that are
86. 3:47in favor of harsher penalties for drug
87. 3:50offenses this proportion is the
88. 3:52parameter p and we're trying to estimate
89. 3:58P how large of a sample size is required
90. 4:01if we wish to estimate P to within 03
91. 4:04with 95%
92. 4:06confidence here's the sample size
93. 4:08formula the sample size must be at least
94. 4:11this
95. 4:12quantity The Zed value corresponding to
96. 4:15a 95% confidence level is
97. 4:201.96 and we want to estimate P to within
98. 4:2303 so m is
99. 4:2703 and we square that Quant
100. 4:32qu what should we use for p star well we
101. 4:35appear to have no estimate of P to use
102. 4:37in the formula and if we have no
103. 4:39reasonable estimate of P we should use
104. 4:42the conservative approach and choose a p
105. 4:44star value of5 so * .5 * 1
106. 4:51-.5 if we carry out this calculation
107. 4:54we'd see that we need a sample size of
108. 4:56at least
109. 4:581,67 .1
110. 5:01individuals but the sample size must be
111. 5:03a whole number to find the minimum
112. 5:05sample size required we need to round up
113. 5:08to the next largest integer
114. 5:111068 we would need a sample size of at
115. 5:13least 1,68 individuals in order to
116. 5:16estimate P the proportion of Ontario
117. 5:19adults that are in favor of harsher
118. 5:20penalties for drug offenses to
119. 5:23within3 with 95%
120. 5:27confidence these sample size
121. 5:29calculations are often used as a rough
122. 5:31approximation to the sample size that is
123. 5:34required we don't necessarily run out
124. 5:36and get a sample size of exactly 1,68
125. 5:40people we often see samples of about
126. 5:43this size in political polls political
127. 5:45polls are often based on a sample of
128. 5:47about 1,000 individuals with a reported
129. 5:5095% margin of error of about
130. 5:5503 here's another example suppose we
131. 5:58wish to estimate the proportion of mice
132. 6:00that would be killed by a certain dose
133. 6:02of a chemotherapy drug suppose also that
134. 6:05previous Studies have shown that
135. 6:06approximately 10% of mice are killed by
136. 6:09this dose of this
137. 6:12drug suppose that we are feeling a
138. 6:15little ambitious and we wish to estimate
139. 6:17P to within
140. 6:18.1 with 99%
141. 6:21confidence how large of a sample size is
142. 6:25required recall that we suspect from
143. 6:27previous studies that P is a prox o
144. 6:29imately
145. 6:31.1 here's the sample size formula we
146. 6:34need a sample size of at least this
147. 6:37quantity for a 99% confidence level to
148. 6:41three decimal places The Zed value is
149. 6:452.5
150. 6:4776 we can find that from software or a
151. 6:50table if we used software to find that
152. 6:53value to more decimal places we'd get a
153. 6:55slightly different answer than we're
154. 6:57going to get but I'm just going to use
155. 6:59three decimal places here there will be
156. 7:01a little bit of rounding
157. 7:04error we want to estimate P to
158. 7:07within1 so m is
159. 7:11.1 and we square that
160. 7:15quantity what should we use for p star
161. 7:18well if we wanted to be conservative we
162. 7:20could again use 0.5 but previous Studies
163. 7:24have shown that P is approximately 0.1
164. 7:27so it's reasonable to use that value in
165. 7:30this formula * .1 * 1us
166. 7:34.1 one could make an argument that we
167. 7:37should still be conservative and use 0.
168. 7:39five but if we feel 0.1 is a reasonable
169. 7:42estimate of P there's probably no need
170. 7:44to be that
171. 7:46conservative if we carry out this
172. 7:48calculation we'd see that we need a
173. 7:50sample size of at least
174. 7:5459721
175. 7:5619.8 mice
176. 7:59if we want the minimum sample size
177. 8:01required then we round up to the next
178. 8:03largest integer the minimum sample size
179. 8:06is
180. 8:1159722 so we'd have to conduct an
181. 8:13experiment and administer the drug to
182. 8:15almost 600,000 mice in order to estimate
183. 8:18P to within .1 with 99%
184. 8:22confidence that brings up the point that
185. 8:24the required sample size may not be
186. 8:26reasonable and it certainly isn't
187. 8:28reasonable here
188. 8:30we're not going to go out and run an
189. 8:31experiment with 600,000 mice just to
190. 8:34estimate P to within
191. 8:360.1 so we may have the desire to
192. 8:39estimate P to within .1 with 99%
193. 8:42confidence but it's simply not possible
194. 8:45from a practical Viewpoint we're going
195. 8:47to have to adjust our study plans or
196. 8:49even abandon them what we wanted to show
197. 8:52we simply cannot
198. 8:54show now let's look at a few plots to
199. 8:57illustrate the effect of the desired
200. 8:59margin of of error and confidence level
201. 9:01on the required sample
202. 9:05size in this first plot I plotted the
203. 9:07minimum required sample size against the
204. 9:10desired 95% margin of error I used the P
205. 9:14star value of 0.5 so this is an
206. 9:16illustration of the values from the
207. 9:18first
208. 9:19example in the first example we had a
209. 9:21desired margin of error of 03 and the
210. 9:24minimum sample size required was 1,68
211. 9:29we needed to sample at least 1,68 people
212. 9:32in order to estimate the population
213. 9:34proportion to within
214. 9:36003 if we wanted to reduce the margin of
215. 9:38error to 01 we'd have to increase the
216. 9:41sample size to almost 10,000 people this
217. 9:45would cost a lot of money and time so
218. 9:47it's usually not worth the
219. 9:50effort the P star value of 0.5 was the
220. 9:53conservative or worst case approach if
221. 9:55we were to use a different value of P
222. 9:57star the minimum sample size would be
223. 10:00less to illustrate suppose we let P star
224. 10:03equal
225. 10:053 this dashed curve is the resulting
226. 10:07minimum sample size curve we see that we
227. 10:10need a little lower sample size if P
228. 10:12star differs from
229. 10:140.5 the sample size formula contains P
230. 10:17star * 1 minus P star so this curve
231. 10:21would be the same if P star equals 3 or
232. 10:24if P star equals
233. 10:277 what happens if we increase the
234. 10:29confidence
235. 10:31level these two green curves represent
236. 10:33the minimum sample size required for a
237. 10:3699% margin of error for a given margin
238. 10:39of error if we wish to have greater
239. 10:41confidence we need a greater sample
240. 10:45size if we want to estimate P to within
241. 10:4801 with 99% confidence we need a sample
242. 10:52size of over 16,000
243. 10:55individuals whatever the confidence
244. 10:57level the required sample size increases
245. 11:00dramatically as the desired margin of
246. 11:02error gets closer to zero so in many
247. 11:05practical situations it may not be
248. 11:07possible to pin down the value of p as
249. 11:09precisely as we would
250. 11:11like and that's a brief introduction to
251. 11:13sample size calculations for estimating
252. 11:16a proportion