Estimation

1
Estimation

• Statistics with Confidence

2
Estimation

• Before we collect our sample, we know

Repeated sampling sample means would stack up in
a normal curve, Centered on the true
population mean,
With a standard error
(measure of dispersion) that depends on



1. population standard deviation
2.
sample size
?
-3z -2z -1z 0z
1z 2z 3z
3
What are they doing?
?
4
Estimation

• But we do not know 1. True Population Mean
• 2. Population Standard Deviation

Repeated sampling sample means would stack up in
a normal curve, Centered on the true
population mean,
With a standard error
(measure of dispersion) that depends on



1. population standard deviation
2.
sample size
?
-3z -2z -1z 0z
1z 2z 3z
5
Estimation

• Will our sample be one of these (accurate)?
• Or one of these (inaccurate)?

?
-3z -2z -1z 0z
1z 2z 3z
6
Estimation

• Which is more likely?
• accurate?
• or inaccurate?

?
-3z -2z -1z 0z
1z 2z 3z
68
95
7
Estimation

• Were most likely to get close to the true
  population mean
• Our samples mean is the best guess of the
  population mean, but it is not precise.

?
-3z -2z -1z 0z
1z 2z 3z
68
95
8
Estimation

• And if we increase our sample size (n)

?
-3z -2z -1z 0z
1z 2z 3z
68
95
9
Estimation

• And if we increase our sample size our sample
  mean is an
• even better estimate of the
• population mean, we are
• more precise!

?
-3 -2 -1 0 1 2 3
-3z -2z
-1z 0z
1z 2z
3z
68
95
10
Estimation

• We know that the standard deviation of this pile
  of samples (standard error) equals the population
  standard deviation (?) divided by the square
• root of the sample size (n).

?
-3 -2 -1 0 1 2 3
68
95
11
Estimation

• But we do not know the population standard
  deviation!
• What is our best guess
• of that?

?
-3 -2 -1 0 1 2 3
68
95
12
Estimation

• Our best guess of the population standard
  deviation is our samples s.d.! On average, this
  s.d. gives population ?.
• In fact, when we calculate that,
• we use n 1 to make our
• estimate larger to reflect
• that dispersion of a sample
• is smaller than a populations.

? (Yi Y)2 s n - 1
Cases in the sample
0 5 10 15 20 25 30 35
0 5 10 15 20 25 30 35
Population Dispersion
Sample Dispersion
13
Estimation

• So now we know that we can use the sample
  standard deviation to stand in for the
  populations standard deviation.
• So we can use the formula
• for standard error with that s
• estimate and get a good estimate s.e
• of the dispersion of the ? n
• sampling distribution.

?
-3 -2 -1 0 1 2 3
68
95
14
Estimation

• Now we know some limits on how far off our sample
  mean is likely to be from the true population
  mean!
• 68 of means will
• be within /- 1 s.e.
• 95 of means will
• be within /- 2 s.e.

s s.e. ? n
?
-3 -2 -1 0 1 2 3
68
95
15
Estimation

• For example, if we took GPAs from a sample of 625
  students and our s was .50
• 68 of means would
• be within /- 1(.02)
• 95 of means would
• be within /- 2(.02)

.5 s.e. ? 625 0.02
?
-3 -2 -1 0 1 2 3
0.02
68
95
16
Estimation

• GPAs from a sample of 625 students with s .50
• If our sample were
• this one,
• our estimate of
• the mean would
• be correct!

.5 s.e. ? 625 0.02
?
-3 -2 -1 0 1 2 3
68
95
17
Estimation

• GPAs from a sample of 625 students with s .50
• But what if it were
• this one?
• Wed be slightly
• wrong, but well within
• /- 2 (.02)
• 95 of samples would be!

.5 s.e. ? 625 0.02
?
-3 -2 -1 0 1 2 3
68
95
18
Estimation

• A samples mean is the best estimate of the
  population mean.
• But what if we base our estimate on this
  erroneous sample?

s s.e. ? n
?
-3 -2 -1 0 1 2 3
68
95
19
Estimation

• Lets create a measuring device with our
  sampling distribution and center it over our
  samples mean.
• Check it Out!
• The true mean falls within
• the 95 bracket.

s s.e. ? n
?
-3 -2 -1 0 1 2 3
68
95
20
Estimation

• What if the sample we collected were this one?
• and we used the measuring device again?
• Check it Out!
• The true mean falls within
• the 95 bracket.

s s.e. ? n
?
-3 -2 -1 0 1 2 3
68
95
21
Estimation

• The sampling distribution allows us to
• 1. Be humble and admit that our sample
• statistic may not be the populations
• and 2. Forms a measuring device
• with which we can determine a range
• where the true population mean
• is likely to fall...
• this is called a confidence interval.

22
Estimation

• If you calculate your sampling distributions
  standard error,
• you can form a device that tells you that
• if your sample mean
• is wrong, there is a
• documented a range in
• which the true
• population mean is likely
• 2Xist.
• Check it Out!
• The true mean falls within
• the 95 bracket.

s s.e. ? n
?
Sample
-3 -2 -1 0 1 2 3
68
95
23
Estimation

• For example, if we took GPAs from a sample of 625
  students and our mean was 2.5 and s.d. was .50
• We make a confidence
• interval (C.I.)by
• Calculating the s.e. (.02)
• and
• Going /- 2 s.e.
• from the mean.

.5 s.e. ? 625 0.02
?2.52
-3 -2 -1 0 1 2 3
68
95
95 C.I. 2.5 /- 2(.02) 2.46 to
2.54 We are 95 confident that the true mean
is in this range!
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Estimation

• Guys This is power!
• Knowing that the spread of 95 of normally
  distributed sample means has outer limits
• We know that if we put these limits around our
  sample mean
• We have defined the range where the population
  mean has a 95 probability of being!

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Estimation

• Our sample statistics provide enough information
  to give us a great estimation (highly educated
  guess) about population statistics.
• We do this without needing to know the population
  meanwithout needing to have a census.

26
Estimation

• Another Example
• Sample of 2,500 with an average income of 28,000
  with a standard deviation of 8,000.
• Provide a 95 C.I. M /- 2 (s.e.)
• s.e. 8,000/?2,500 160
• 2 160 320
• C.I. 28,000 /- 320
• C.I. gtgtgt 27,680 to 28,320

s s.e. ? n
27
Estimation

• Another Example
• Sample of 2,500 with an average income of 28,000
  with a standard deviation of 8,000.
• Provide a 95 C.I. M /- 2 (s.e.)
• s.e. 8,000/?2,500 160
• 2 160 320
• C.I. 28,000 /- 320
• C.I. gtgtgt 27,680 to 28,320

We are 95 confident that the true mean falls
from 27,680 up to 28,320
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Estimation

• NO WAIT! Were wrong!
• Technically speaking, on a normal curve, 95 of
  cases fall between /- 1.96 standard deviations
  rather than 2.
• (Check your books table.)
• Empirical Rule vs. Actuality
• 68 1z 0.99z
• 95 2z 1.96z
• 99.9973 3z 3z

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Estimation

• Another Example
• Sample of 2,500 with an average income of 28,000
  with a standard deviation of 8,000.
• Provide a 95 C.I. M /- 1.96 (s.e.)
• s.e. 8,000/?2,500 160
• 1.96 160 313.6
• C.I. 28,000 /- 313.6
• C.I. gtgtgt 27,686.4 to 28,313.6

We are 95 confident that the population mean
falls between 27,686.4 and 28,313.6
30
Estimation

• Another Example
• Sample of 2,500 with an average income of 28,000
  with a standard deviation of 8,000.
• What if we want a 99 confidence interval, What
  z do we use?
• Check the table in your book!

31
Estimation

• Another Example
• Sample of 2,500 with an average income of 28,000
  with a standard deviation of 8,000.
• What if we want a 99 confidence interval?
• 99 fall between /- 2.58 zs

32
Estimation

• Another Example
• Sample of 2,500 with an average income of 28,000
  with a standard deviation of 8,000.
• What if we want a 99 confidence interval?
• s.e. 8,000/?2,500 160
• 2.58 160 412.8
• C.I. 28,000 /- 412.8
• CI gtgtgt 27,587.2 to 28,412.8
• We are 99 confident that the population mean
  falls between these values.
• Why did the interval get wider than 95 CIs
  which was 27,686.4 to 28,313.6???

33
Estimation

• 99 CI gtgtgt 27,587.2 to 28,412.8
• Why did the interval get wider than 95 CIs
  which was
• 27,686.4 to
  28,313.6???

M
-3 -2 -1 0 1 2 3
68
99
95
34
Estimation

• Lets recap We can say that 95 of the sample
  means in repeated sampling will always be in the
  range marked by -1.96 over to 1.96 standard
  errors.

Self-esteem 15 20 25 30 35
40
1.96
Z-3 -2 -1 0 1 2 3
-1.96
95 Range
z -3 -2 -1 0 1 2 3
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Estimation

• And remember If we dont know the true
  population mean, 95 of the time a 95 confidence
  interval would contain the true population mean!

Self-esteem 15 20 25 30 35
40
95 Ranges for different samples.
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Estimation

• If we want that range to contain the true
  population mean 99 of the time (99 confidence
  interval) we just construct a wider interval,
  corresponding with 2.58 zs.

Self-esteem 15 20 25 30 35
40
99 Ranges for different samples, overlaying 95
intervals.
37
Estimation
1.96z
The sampling distributions standard error is a
measuring stick that we can use to indicate the
range of a specified middle percentage of sample
means in repeated sampling.
95
1z
68
3z
99.99
25
-3 -1.96 -1 0 1 1.96 3
68
95
99.99
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Estimation

• Another Confidence Interval Example
• I collected a sample of 2,500 with an average
  self-esteem score of 28 with a standard deviation
  of 8.
• What if we want a 99 confidence interval? CI
  Mean /- z s.e.
• Find the standard error of the sampling
  distribution
• s.d. / ?n 8/50 0.16
• Build the width of the Interval. 99 corresponds
  with a z of 2.58.
• 2.58 0.16 0.41
• Insert the mean to build the interval
• 99 C.I. 28 /- 0.41
• The interval 27.59 to 28.41
• We are 99 confident that the population mean
  falls between these values.

39
Estimation

• And if we wanted a 95 Confidence Interval
  instead?
• I collected a sample of 2,500 with an average
  self-esteem score of 28 with a standard deviation
  of 8.
• What if we want a 99 confidence interval? CI
  Mean /- z s.e.
• Find the standard error of the sampling
  distribution
• s.d. / ?n 8/50 0.16
• Build the width of the Interval. 99 corresponds
  with a z of 2.58.
• 2.58 0.16 0.41
• Insert the mean to build the interval
• 99 C.I. 28 /- 0.41
• The interval 27.59 to 28.41
• We are 99 confident that the population mean
  falls between these values.

95
X
95
1.96
X
X
X
X
0.31
1.96
X
X
95
0.31
X
X
27.69 to 28.31
X
95
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Estimation

• By centering my sampling distributions /- 1.96z
  range around my samples mean...
• I can identify a range that, if my sample is one
  of the middle 95, would contain the populations
  mean.
• Or
• I have a 95 chance that the populations mean is
  somewhere in that range.

41
Estimation

• By centering my sampling distributions /- 1.96z
  range around my samples mean...
• I can identify a range that, if my sample is one
  of the middle 95, would contain the populations
  mean.
• Or
• I have a 95 chance that the populations mean is
  somewhere in that range.

X
2.58z
X
99
99
X