Estimating the Population Mean
Income of Lexus Owners
• Sample Mean + Margin of Error
• Called a Confidence Interval
• To Compute Margin of Error, One of Two Conditions
Must Be True:
• The Distribution of the Population of Incomes Must Be
Normal, or
• The Distribution of Sample Means Must Be Normal.
A Side-Trip Before Constructing
Confidence Intervals




What is a Population Distribution?
What is a Distribution of the Sample Mean?
How Does Distribution of Sample Mean Differ
From a Population Distribution?
What is the Central Limit Theorem?
Assume Small Population of
Lexus Owners’ Incomes (N = 200)
190
149
187
361
97
76
302
234
371
276
165
162
330
174
247
272
349
275
256
199
105
89
182
91
135
353
263
314
353
295
316
316
242
101
287
153
361
353
354
185
254
85
348
287
94
105
127
317
346
250
284
312
337
368
195
305
253
211
235
206
345
124
215
320
183
152
196
154
238
121
156
278
207
161
108
285
242
125
115
174
340
141
203
367
221
308
241
128
225
343
346
302
140
235
344
333
365
94
100
138
363
80
166
215
228
279
288
335
277
188
87
360
333
197
191
76
220
77
248
297
79
161
368
165
187
318
242
109
222
135
288
261
159
374
104
279
152
237
324
232
197
178
317
300
371
292
129
93
119
137
211
292
286
343
308
354
320
260
95
344
334
241
372
180
87
101
366
303
86
175
230
365
287
318
310
88
224
223
156
256
117
182
152
308
144
115
281
297
314
173
152
186
188
348
275
230
330
249
285
232
Distribution of N = 200 Incomes
Mean
30
75 125 175 225 275 325
Constructing a Distribution of
Samples of Size 5 from N = 200 Owners
Obs 1
362
80
166
214
228
278
288
335
277
188
86
359
332
Obs 2
Obs 3
79
160
367
165
187
317
241
109
221
135
287
260
159
197
177
316
300
370
292
129
92
118
136
211
291
285
Obs 4
333
241
372
180
87
100
366
303
86
175
229
365
287
x
Obs 5
Mean
116
182
151
307
144
114
281
296
313
172
151
185
187
217.4
168.0
274.4
233.2
203.2
220.2
261.0
227.0
203.0
161.2
192.8
292.0
250.0
Distribution of Sample Mean
Incomes (Column #7)
25
21
Estimated Std. Error
20
15
8
10
6
3
5
2
0
125
175
225
275
325
Distribution of Sample Means
Near Normal!
( X)
Central Limit Theorem

Even if Distribution of Population is Not Normal, Distribution of
Sample Mean Will Be Near Normal Provided You Select Sample
of Five or Ten or Greater From the Population.
For a Sample Sizes of 30 or More, Dist. of the Sample Mean Will
Be Normal, with
 mean of sample means = population mean, and
 standard error = [population deviation] / [sqrt(n)]

Thus Can Use Expression:

X  MOE
Why Does Central Limit Theorem
Work?

As Sample Size Increases:

Most Sample Means will be Close to
Population Mean,


Some Sample Means will be Either Relatively Far
Above or Below Population Mean.
A Few Sample Means will be Either Very Far
Above or Below Population Mean.
Impact of Side-Trip on MOE
MOE = (Confidenc e factor)  Spread in Xs




Determine Confidence, or Reliability, Factor.
Distribution of Sample Mean Normal from Central
Limit Theorem.
Use a “Normal-Like Table” to Obtain Confidence
Factor.
Determine Spread in Sample Means (Without
Taking Repeated Samples)
Drawing Conclusions about a Pop.
Mean Using a Sample Mean
Select Simple Random Sample
Compute Sample Mean and
Std. Dev. For n < 10, Sample Bell-Shaped?
For n >10 CLT Ensures Dist of x Normal
x  MOE
Draw Conclusion about
Population Mean, m
Federal Aid Problem

Suppose a census tract with 5000
families is eligible for aid under program
HR-247 if average income of families of
4 is between $7500 and $8500 (those
lower than 7500 are eligible in a
different program). A random sample of
12 families yields data on the next
page.
Federal Aid Study Calculations
Representative Sample
7,300 7,700 8,100 8,400
7,800 8,300 8,500 7,600
7,400 7,800 8,300 8,600
x  $7,983
s  $441 
( 7300  7983)2 ..(8600  7983)2
12  1
x  MOE
7,983  MOE
Estimated Standard Error




Measures Variation Among the Sample Means If We
Took Repeated Samples.
But We Only Have One Sample! How Can We
Compute Estimated Standard Error?
Based on Constructing Distribution of Sample Mean
Slide, Will Estimated Standard Error Be Smaller or
Larger Than Sample Standard Deviation (s)?
Estimated Std. Error ______ than s.
Estimated Standard Error Expression
sample standard deviation
sample size
For Federal
Aid Study
s

n
Confidence Factor for MOE:
Appendix 5
Df = n-1
2-Sided
90%
2-Sided
95%
2-Sided
99%
1.86
2.306
3.355
11
1.796
2.201
3.106
14
1.761
2.145
2.977
17
1.740
2.110
2.898
8
Can Use t-Table Provided Distribution
of Sample Mean is Normal
95% Confidence Interval
X  Confidence  Est. S tan dard Error
441
7, 983  ( 2. 201) 
12
$7, 703    $8, 263
Interpretation of Confidence
Interval
 95% Confident that Interval $7,983 + $280
Contains Unknown Population (Not
Sample) Mean Income.
 If We Selected 1,000 Samples of Size 12 and
Constructed 1,000 Confidence Intervals,
about 950 Would Contain Unknown
Population Mean and 50 Would Not.
 So Is Tract Eligible for Aid???
Would Tract Be Eligible?
X  Conf Factor  Spread in Means

Situation A: 7,700+ 150
7,550    7,850

Situation B: 8,250+ 150
8,100    8,400

Situation C: 8,050+ 150
7,900    8,200
Width versus Meaningfulness of
Two-Sided Confidence Intervals
For n = 1000
60 % CI
WIDTH
s
MOE  0.842 
n
CONFIDENCE
LOW
Ideal:
99 % CI
MOE 
HIGH
_________ Level of Confidence and
_________ Confidence Interval .
How Obtain?
Chapter Summary



Why Must We Estimate Population
Mean?
Why Would You Want to Reduce MOE?
How Can MOE Be Reduced Without
Lowering Confidence Level?