Confidence Interval Estimation for the Population Proportion — Transcript

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1. 0:00welcome to this tutorial on interval
2. 0:02estimation for the population proportion
3. 0:05just as we have created confidence
4. 0:07intervals for the population mean with
5. 0:10both Sigma known and sigma unknown we
6. 0:13can also construct a confidence interval
7. 0:15estimate for the population
8. 0:18proportion the general form of an
9. 0:20interval estimate is the sample
10. 0:22statistic plus or minus a margin of
11. 0:24error for a proportion that would be as
12. 0:28shown here P bar plus or minus some
13. 0:31margin of error we have seen in previous
14. 0:34tutorials that the sampling distribution
15. 0:36of P bar can be approximated by a normal
16. 0:40distribution whenever n * p is greater
17. 0:44than or equal to 5 and n * 1 minus p is
18. 0:48also greater than or equal to 5 you can
19. 0:50refer back to the tutorial on sampling
20. 0:52distributions if you need a refresher on
21. 0:55the topic but for this tutorial you need
22. 0:58to remember that the mean of the
23. 1:01sampling distribution of P bar is the
24. 1:04population proportion p and the standard
25. 1:08error of P bar is shown as Sigma
26. 1:12subscript P bar and that is equal to the
27. 1:14square root of P * 1us P / little n the
28. 1:19sample size this standard Arrow will now
29. 1:22be used in the formula for computing a
30. 1:24confidence interval estimate for
31. 1:27p here is the formula for calculating
32. 1:30the confidence interval estimation for a
33. 1:32proportion we can see that the standard
34. 1:34error is used as an integral part of the
35. 1:36formula we also see that we will be
36. 1:39using the Z table with Alpha divided in
37. 1:42half since the sampling distribution of
38. 1:44P bar is normally distributed so to
39. 1:47calculate the interval we have the
40. 1:48formula shown here P bar which is a
41. 1:51sample proportion plus and minus Z of
42. 1:54alpha / half so that is the value we
43. 1:57look up in the Z table times the
44. 2:00standard error for p bar which is the
45. 2:02square < TK of P * 1us P bar / little n
46. 2:07the original formula for the standard
47. 2:09error of P bar shown on the previous
48. 2:10slide actually uses P not P bar but
49. 2:14since we don't know what P the
50. 2:16population proportion is we use P bar as
51. 2:19a point estimator for p let's see how
52. 2:22this formula might work in an example
53. 2:25let's say that 200 fans were surveyed at
54. 2:28the stadium to see how many order at
55. 2:30least one hot dog and we find that 130
56. 2:33of these 200 fans say they ordered one
57. 2:36at least one hot dog then P bar is 130 /
58. 2:41200 or 65 that is 65% of the people we
59. 2:47asked 130 out of 200 said that they
60. 2:49ordered at least one hot dog n the
61. 2:52sample size is
62. 2:54200 therefore the standard error for p
63. 2:57bar is calculated as the square root of
64. 3:0065 * 1
65. 3:03-65 / 200 and we get
66. 3:090337 now that we have the standard error
67. 3:12we are ready to use the formula for
68. 3:14constructing a confidence interval on
69. 3:17P here is the general formula again we
70. 3:20have P bar that is 65 we have n that is
71. 3:25200 and we have the standard error which
72. 3:28is 033 7 so what is left is to determine
73. 3:32Z of alpha / in
74. 3:36half now let's say we want to construct
75. 3:39a 99% confidence interval on the true
76. 3:41population proportion P here we can see
77. 3:44the distribution marked off with the
78. 3:46area of 099 around the middle of the
79. 3:49distribution this N9 corresponds to a
80. 3:5399% confidence interval first we have to
81. 3:56determine our level of significance
82. 3:58Alpha which would be the tail areas now
83. 4:01if we are drawing a 99% confidence
84. 4:03interval then Alpha is 1 minus our level
85. 4:07of confidence so 1
86. 4:09-99 is 01 so Alpha is
87. 4:1301 this is the area in both the upper
88. 4:16tail and lower area Tails so we split
89. 4:19Alpha and half with half in the upper
90. 4:20tail and half in the lower tail and
91. 4:22Alpha divided in half is
92. 4:2605 now we are ready to look up 00 05 in
93. 4:30the netive Z
94. 4:33table looking in the middle of the table
95. 4:36we find two numbers are close to
96. 4:3905 we have
97. 4:420.49 and
98. 4:440.51 and obviously 05 would be right in
99. 4:48the middle of these two numbers looking
100. 4:50to the left we find
101. 4:52-2.5 and looking up we find that we are
102. 4:55in between 007 and 08 so we get a z
103. 5:00value of - 2.57 5 which is between -
104. 5:042.57 and -
105. 5:072.58 for this example we will use a zv
106. 5:10value of plus or minus
107. 5:132575 if you watched my previous tutorial
108. 5:16on common Z values we used a value of
109. 5:192576 since that is the value we would
110. 5:22get without a rounding error or if we
111. 5:24used Excel to compute the zv value but
112. 5:27since we manually looked it up in this
113. 5:29table let's use the table
114. 5:31value okay so now getting back to
115. 5:34calculating the interval let's start to
116. 5:36plug in the numbers into the formula P
117. 5:39bar is 65 so we get 65 plus and minus
118. 5:472575 the zv value that we looked up in
119. 5:49the table times the square < TK of 65 *
120. 5:541 - 65 or. 35 / 200 and we we get 65
121. 6:01plus andus
122. 6:032575 *
123. 6:060337 which is 65 plus and minus
124. 6:12868 which is an interval of between.
125. 6:155632 and
126. 6:187368 this represents the interval within
127. 6:22which we are 99% confident that the true
128. 6:25population proportion exists plotted on
129. 6:28the normal distribution curve the
130. 6:30interval would look like
131. 6:32this this is the normal distribution
132. 6:34curve with P the population proportion
133. 6:37marked off in the middle and a 99%
134. 6:40confidence interval drawn around it we
135. 6:42can see the lower limit is marked at0
136. 6:455632 and the upper limit is marked at
137. 6:497368 when we say this is a 99%
138. 6:52confidence interval we are saying that
139. 6:54based on this sample of 200 we are 99%
140. 6:58confident that the true portion of fans
141. 7:00ordering hot dogs at the stadium is
142. 7:02within the interval between and
143. 7:04including 5632 and
144. 7:097368 that concludes this tutorial on
145. 7:11confidence interval estimation for
146. 7:13population proportions I hope you
147. 7:15enjoyed this tutorial and I hope you
148. 7:17learned
149. 7:27something