The Sampling Distribution of the Sample Mean — Transcript

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1. 0:01Let's talk about the sampling distribution of the sample mean X bar.
2. 0:06We're going to let X_1 through X_n represent n independent observations
3. 0:10from a population with mean mu and standard deviation sigma.
4. 0:14In simpler terms, we're drawing a random sample of n observations from this distribution.
5. 0:20And we're going to let X bar represent the mean of these n observations.
6. 0:25X bar is a sample statistic, representing the mean of the sample.
7. 0:28Before we draw our sample we can think of X bar as a random variable.
8. 0:32The value of X bar will depend on the sample we draw,
9. 0:35and if we were to repeatedly sample from the population,
10. 0:38then the value of the sample mean X bar would vary from sample to sample.
11. 0:43So the sample mean X bar is a random variable with a probability distribution,
12. 0:48and we call that probability distribution the sampling distribution of X bar.
13. 0:54Here are two important characteristics of the sampling distribution of X bar.
14. 0:58I'm stating them without proof here,
15. 1:01but I do have a video in which I derive these properties.
16. 1:03Mu sub X bar is notation for the mean of the sampling distribution of X bar,
17. 1:09and that is equal to the mean of the population from which we are sampling.
18. 1:13Or in other words
19. 1:15the expectation of our random variable X bar is equal to mu,
20. 1:21the mean of the population from which we are sampling.
21. 1:23The standard deviation of the sampling distribution of X bar,
22. 1:28which we'll represent by sigma with a subscript of X bar,
23. 1:31is equal to the standard deviation of the population from which we are sampling,
24. 1:35divided by the square root of the sample size.
25. 1:38And the standard deviation is just the square root of the variance,
26. 1:43and so the variance of X bar is equal to that quantity squared,
27. 1:47or in other words, sigma squared over n.
28. 1:51If the population from which we are sampling is normally distributed,
29. 1:57then X bar is also normally distributed. And so in summary,
30. 2:01if we are sampling n observations from a normally distributed population,
31. 2:05X bar is distributed normally with
32. 2:08a mean of mu and a variance of sigma squared over n.
33. 2:12Let's see what that looks like.
34. 2:17Suppose we are sampling from this distribution,
35. 2:21which is a normal distribution with a mean of mu and a standard deviation of sigma.
36. 2:25This distribution given in red is the sampling distribution
37. 2:29of the sample mean X bar for sample size of 2.
38. 2:34The distribution in green is the sampling distribution of X bar for a sample size of 4,
39. 2:39and the distribution in orange is the sampling distribution of X bar for a sample size of 8.
40. 2:46In all of these situations,
41. 2:48the mean of the sampling distribution is always equal to the mean of the population
42. 2:51from which we are sampling,
43. 2:53but the standard deviation is smaller.
44. 2:57Here, the standard deviation of the population from which we are sampling,
45. 3:01represented by the white curve is sigma.
46. 3:04When the sample sizes is 2,
47. 3:07the standard deviation of the sampling distribution of X bar,
48. 3:11which we represent by sigma sub X bar, is equal to sigma over the square root of 2.
49. 3:16and when the sample size is 4 the sampling distribution of X bar
50. 3:21has a standard deviation
51. 3:25of sigma over the square root of 4.
52. 3:29And when the sample size is 8, like in the orange curve, the standard deviation
53. 3:33of the sampling distribution of X bar is equal to sigma over the square root of 8.
54. 3:38So the standard deviation of the sampling distribution of X bar decreases as the sample size increases.
55. 3:46But it decreases by a factor of the square root of the sample size,
56. 3:49so if we wanted to cut the standard deviation of the sampling distribution in half
57. 3:54we would need to quadruple the sample size.
58. 3:58A main reason we like large sample sizes in statistics
59. 4:01is that for a larger sample size the distribution of a statistic
60. 4:04is going to be more tightly grouped around the parameter it estimates,
61. 4:08and so we're going to be able to estimate that parameter with greater precision.
62. 4:15In probability calculations we will standardize in the usual way.
63. 4:18If we are sampling a single value from a normally distributed population,
64. 4:22then that single value, X say,
65. 4:26has a normal distribution with a mean of mu and a variance of sigma squared.
66. 4:34To standardize, we subtract the mean and divide by the standard deviation,
67. 4:37so if we let the random variables Z equal
68. 4:41X minus its mean of mu and divide by its standard deviation of sigma,
69. 4:46then Z has the standard normal distribution.
70. 4:52For probability calculations involving the mean of n observations,
71. 4:54X bar is distributed normally with a mean of mu
72. 5:01and a variance of sigma squared over n,
73. 5:04and so to standardize here, we're going to let the random variable
74. 5:10Z equal X bar minus its mean,
75. 5:13which is simply mu, and divided by the standard deviation of X bar,
76. 5:17which is sigma over the square root of n.
77. 5:22And if we do that
78. 5:25then the random variables Z is going to have the standard normal distribution.
79. 5:30Let's look at an example of this type of probability calculation.
80. 5:34The amount of protein in a quarter pound patty,
81. 5:38or about 113 grams of lean beef, is approximately normally distributed with
82. 5:42a mean of 21.4 grams and a standard deviation of 1.9 grams.
83. 5:46This is based on information from the USDA nutrient database.
84. 5:52Suppose we want to know the probability
85. 5:54a single randomly selected patty has at least 23.0 grams of protein.
86. 5:58Here again are the mean and standard deviation for the population,
87. 6:02and I've plotted in the distribution of the amount of protein in a single burger here.
88. 6:08And if we let the random variable X
89. 6:10represent the amount of protein in a randomly selected burger,
90. 6:13then here we want to know the probability that the
91. 6:17random variable X takes on a value that's at least 23.0.
92. 6:21And that is simply
93. 6:24the area to the right of 23.0 under this distribution.
94. 6:30This area right here.
95. 6:32To find that, we can standardize by subtracting the mean and dividing by the standard deviation.
96. 6:38And this is equal to the probability that the random variable Z,
97. 6:43where Z has the standard normal distribution, takes on a value that's
98. 6:46bigger than or equal to 23.0
99. 6:49minus the mean of 21.4
100. 6:52divided by the standard deviation of 1.9.
101. 6:56And that's simply equal to the probability that the random variable Z
102. 7:01takes on a value that's bigger than or equal to 0.842.
103. 7:07And we draw out our standard normal curve,
104. 7:080 in the middle, out here somewhere is 0.842,
105. 7:13and we need this area here.
106. 7:16If we go to software or a standard normal table
107. 7:19we can find that, to three decimal places, this is 0.200.
108. 7:25So that's the probability we're looking for.
109. 7:30That last question involved the amount of protein in a single patty,
110. 7:33and here the question changes a little bit.
111. 7:35What is the probability the mean amount of protein
112. 7:38in 4 randomly selected patties is at least 23 grams?
113. 7:41Here again in white is the distribution of protein in a single patty.
114. 7:46But in this question we're interested in the probability based on
115. 7:49the mean amount of protein in 4 patties.
116. 7:52And I'm going to draw in this curve in green, which is the sampling distribution of the sample mean in this scenario.
117. 7:59This sampling distribution has the same mean
118. 8:02as the original population of 21.4, but the standard deviation
119. 8:07of this sampling distribution in green is sigma of 1.9 over the square root of n,
120. 8:13which is the square root of 4. And in this question we need to find the probability
121. 8:20that the random variable X bar takes on a value that's bigger than or equal to 23.0
122. 8:25and that is going to be the
123. 8:27area to the right of 23.0 under the green curve,
124. 8:31under the sampling distribution of X bar. That area right there.
125. 8:36And to find that probability, we can again standardize.
126. 8:42But we have to remember that when we're talking about the mean of n observations,
127. 8:46to standardize we're going to let
128. 8:48Z equal X bar minus mu over sigma divided by the square root of n.
129. 8:53So here this probability we're interested in, we're going to turn this into the probability
130. 8:57that Z is bigger than or equal to 23.0 minus 21.4
131. 9:06all over 1.9 divided by the square root of 4.
132. 9:11And this works out to the probability
133. 9:15that Z takes on a value that's bigger than or equal to 1.684.
134. 9:20And if we go to software or a standard normal table,
135. 9:24we can find that that probability, to three decimal places, is 0.046.
136. 9:31Here the distribution of the sample mean was approximately normal
137. 9:35because we were sampling from an approximately normally distributed population.
138. 9:39But there is a very important concept in statistics that helps us calculate
139. 9:43probabilities involving the sample mean
140. 9:45when we are sampling from distributions that are not normal,
141. 9:49and let's take a quick look at that.
142. 9:53The gist of the central limit theorem
143. 9:55is that if we are sampling from a distribution that is not normal,
144. 9:58the sample mean will still be approximately normal,
145. 10:01provided we have a large sample size.
146. 10:04The central limit theorem is an extremely important concept in statistics,
147. 10:08and I'm just introducing it very briefly here, but I have another video that goes into it in greater detail.
148. 10:19We will use characteristics of the sampling distribution of X bar
149. 10:23to help us answer questions like: how close is X bar likely to be to mu?
150. 10:27In the calculation examples in this video,
151. 10:30we happened to know the value of mu. But that's not usually going to be the case.
152. 10:34In practical situations we don't typically know mu,
153. 10:38it is an unknown value that we're trying to estimate. We're going to use our
154. 10:43knowledge of the characteristics of the sampling distribution of X bar
155. 10:46to help a state how close X bar is likely to be to this unknown value of mu.
156. 10:51This will help us quantify the uncertainty that is always present in this type of estimation.
157. 10:56The notion of a sampling distribution of a statistic plays a very important role in statistical inference.
158. 11:03One small note to finish.
159. 11:06In this video it was assumed that we were sampling from an infinite populatin,
160. 11:09or that we were sampling only a very small fraction of a finite population.
161. 11:13This is typically the case in practice.
162. 11:18In the event we are sampling a larger fraction from a finite population,
163. 11:22the standard deviation of the sampling distribution of the sample mean
164. 11:25changes a little bit from what was given here,
165. 11:29and we need to use something called the finite population correction factor.
166. 11:32But the details of that are another talk for another day.