Estimation of a Population Mean, 
100(1-)% Confidence Interval for a Population Mean using a Large Samplen  
x  z / 2
where

n
x is the sample mean in a simple random sample and
 is the population standard
deviation (if  is unknown, s may be used instead as an approximation).
In-Class Exercise
1) Suppose 40 teachers from Washington have been randomly selected and their salaries recorded. The
resulting sample average was $29,000 and the standard deviation was $2,000.
a. Construct a 90, 95, and 99% confidence interval for , the mean salary for the population of teachers
in Washington. What do you learn about the width of the intervals as you increase the confidence
level?
b. Suppose our sample size is 80 instead of 60. Construct a 95% confidence interval for .
c. Now suppose our sample size is 100. Construct a 95% confidence interval for What do you
learn as you increase the sample size?
2) Weights of women in one age group are normally distributed with a standard deviation σ of 21
lb. A researcher wishes to estimate the mean weight of all women in this age group. Find how
large a sample must be drawn in order to estimate the population mean within 2 pounds with
95% confidence.
3) Suppose that a random sample of 50 bottles of a particular brand of cough medicine is
selected and the alcohol content of each bottle is determined. Let  denote the average alcohol
content for the population of all bottles of the brand under study. Suppose that the sample of 50
results in a 95% confidence interval for  of (7.8, 9.4).
a. Would a 90% confidence interval have been narrower or wider than the given interval?
Explain your answer.
b. There is a correct and wrong interpretation of the confidence interval above. Indicate
which of the two interpretations below is correct and which is wrong.
1. There is a 95% chance that  is between 7.8 and 9.4.
2. We are 95% confident that  is between 7.8 and 9.4. The 95% confidence level
describes our confidence in the procedure used to determine the interval. If we take
sample after sample (each of size 50) from the population and use each one separately to
compute a 95% confidence interval, in the long run 95% of these intervals will capture 
Confidence Intervals calculated from a Minitab simulation for
100 Random Samples(each of size n = 50)
from a normal population with
and =4
Variable
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
C11
C12
C13
C14
C15
C16
C17
C18
C19
C20
C21
C22
C23
C24
C25
C26
C27
C28
C29
C30
C31
C32
C33
C34
C35
C36
C37
C38
C39
C40
C41
C42
C43
C44
C45
C46
C47
C48
C49
C50
C51
C52
C53
C54
N
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
Mean
60.257
61.038
59.365
60.059
60.437
60.308
60.158
59.630
59.175
60.298
59.784
60.429
60.413
60.201
59.197
60.220
60.501
59.448
60.904
59.590
60.657
60.706
58.927
60.654
60.536
59.912
60.862
60.170
60.382
59.496
60.496
59.893
61.075
59.600
59.543
59.848
59.889
59.807
60.286
59.863
60.594
59.938
60.336
60.047
59.836
59.581
60.127
60.811
60.032
59.979
58.791
61.612
59.687
59.999
StDev
3.933
3.943
3.674
4.301
3.292
3.849
3.715
4.421
3.496
3.347
3.274
4.018
3.875
4.068
4.028
4.290
3.425
4.206
4.310
3.642
3.857
4.273
3.557
3.739
3.707
3.886
3.633
3.758
3.983
4.337
4.161
4.234
3.673
4.164
3.221
3.357
3.554
4.922
4.811
4.364
3.667
3.809
4.427
4.158
4.215
4.291
3.706
3.882
3.788
3.253
3.697
4.202
3.382
4.087
SE Mean
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
95.0% CI
59.148, 61.366)
59.929, 62.147)
58.256, 60.473)
58.950, 61.167)
59.328, 61.546)
59.200, 61.417)
59.049, 61.267)
58.522, 60.739)
58.066, 60.283)
59.190, 61.407)
58.675, 60.892)
59.320, 61.537)
59.304, 61.521)
59.092, 61.310)
58.089, 60.306)
59.111, 61.329)
59.392, 61.610)
58.339, 60.557)
59.795, 62.013)
58.481, 60.699)
59.549, 61.766)
59.597, 61.815)
57.818, 60.036)
59.545, 61.763)
59.428, 61.645)
58.803, 61.020)
59.753, 61.971)
59.061, 61.279)
59.274, 61.491)
58.387, 60.605)
59.387, 61.605)
58.784, 61.002)
59.967, 62.184)
58.491, 60.709)
58.434, 60.652)
58.739, 60.957)
58.781, 60.998)
58.698, 60.916)
59.177, 61.395)
58.754, 60.972)
59.485, 61.702)
58.829, 61.047)
59.227, 61.445)
58.938, 61.156)
58.727, 60.945)
58.472, 60.690)
59.019, 61.236)
59.703, 61.920)
58.923, 61.140)
58.870, 61.088)
57.683, 59.900)
60.503, 62.721)
58.579, 60.796)
58.890, 61.107)
C55
C56
C57
C58
C59
C60
C61
C62
C63
C64
C65
C66
C67
C68
C69
C70
C71
C72
C73
C74
C75
C76
C77
C78
C79
C80
C81
C82
C83
C84
C85
C86
C87
C88
C89
C90
C91
C92
C93
C94
C95
C96
C97
C98
C99
C100
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
60.863
60.369
60.081
58.628
59.147
60.594
59.019
59.931
61.032
60.475
60.538
60.719
60.063
59.664
60.856
60.464
59.386
60.118
60.211
59.769
60.860
60.982
59.524
60.282
59.893
59.502
59.593
59.706
59.061
59.590
59.697
59.263
60.749
58.765
60.702
59.130
60.045
60.253
59.123
60.460
59.264
59.861
59.154
60.478
59.661
59.889
3.445
3.697
3.556
3.533
4.223
4.381
4.548
4.153
3.842
3.739
4.227
3.477
3.546
4.540
3.968
3.697
3.538
4.388
3.643
4.421
3.826
3.982
4.035
3.725
4.924
3.469
3.948
3.702
4.070
4.192
4.129
4.213
3.488
4.254
3.496
4.160
4.300
3.334
4.075
4.247
3.816
4.342
3.760
3.958
4.128
3.966
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
0.566
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
59.754,
59.261,
58.972,
57.519,
58.039,
59.485,
57.910,
58.822,
59.924,
59.367,
59.430,
59.611,
58.954,
58.556,
59.747,
59.355,
58.277,
59.010,
59.102,
58.661,
59.751,
59.874,
58.415,
59.173,
58.784,
58.393,
58.484,
58.598,
57.952,
58.481,
58.589,
58.154,
59.640,
57.657,
59.593,
58.022,
58.936,
59.144,
58.014,
59.352,
58.155,
58.752,
58.046,
59.369,
58.552,
58.780,
61.971)
61.478)
61.189)
59.737)
60.256)
61.703)
60.128)
61.040)
62.141)
61.584)
61.647)
61.828)
61.172)
60.773)
61.964)
61.572)
60.494)
61.227)
61.319)
60.878)
61.969)
62.091)
60.632)
61.390)
61.002)
60.611)
60.702)
60.815)
60.169)
60.699)
60.806)
60.371)
61.857)
59.874)
61.811)
60.239)
61.154)
61.361)
60.231)
61.569)
60.373)
60.969)
60.263)
61.587)
60.769)
60.998)
Confidence Interval for 
Using a Small Sample ( n  30 )
x  t / 2,n 1 
s
n
Assumptions:
1. A simple random sample is selected from the population
2. The population distribution is approximately normal.
t Distribution
When x is the mean of a random sample of size n from a normal distribution with mean  , the
random variable
t
x
s
n
has a probability distribution called a t distribution with n – 1 degrees of freedom (df).
Properties of t Distributions
Let tv denote the density function curve for v df.
1. Each tv curve is bell-shaped and centered at 0.
2. Each tv curve is more spread out than the standard normal (z) curve.
3. As    , the sequence of tv curves approaches the standard normal curve (so the z curve is
often called the t curve with df   )
Historical Perspective
The distribution of the t statistic in repeated sampling was discovered by W. S. Gosset, a chemist
in the Guinness brewery in Ireland, who published his discovery in 1908 under the pen name of
Student.
Example A survey was conducted to estimate , the mean salary of middle-level bank
executives. A random sample of 15 executives yielded the following yearly salaries (in units of
$1000):
88
69
121
82
75
80
39
84
52
72
102
115
95
106
78
Find a 98% confidence interval for . Interpret the interval!
Assessing Reasonableness of Normality Assumption
Histogram of salary
Probability Plot of salary
Normal
5
99
Mean
StDev
N
AD
P-Value
95
90
83.87
22.09
15
0.193
0.873
4
Frequency
Percent
80
70
60
50
40
30
3
2
20
10
1
5
1
50
75
100
125
150
0
40
60
salary
80
salary
100
Constructing Confidence Interval
Variable
C19
N
15
Mean
83.8667
StDev
22.0871
SE Mean
5.7029
98% CI
(68.8996, 98.8338)
120