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Population
Sample
Statistic:
Parameter:
Proportion
Proportion p
Count
Count
Mean 
Median
Mean
x
Median
pˆ
Estimate population proportion,
with a confidence interval, from
data of a random sample.
p
proportion pˆ
pˆ
pˆ
Sample
Proportion = p
Population
pˆ
2
p1  p 
n
2
p1  p 
n
p
proportion pˆ
Sample
ˆ
There is 95% chance that p
will fall inside the interval
p2
p1  p 
n
Proportion = p
Population
ND
2
p1  p 
n
2
p1  p 
n
p
proportion pˆ
Sample
There is 95% chance that p
will fall inside the interval
ˆ 2
p
Proportion = p
Population
p1  p 
n
2
pˆ 1  pˆ 
n
2
pˆ 1  pˆ 
n
p
proportion pˆ
Sample
There is 95% chance that p
will fall inside the interval
ˆ 2
p
Proportion = p
Population
ˆ 1  p
ˆ
p
n
Open the Fathom file
‘estimate1.ftm’
The proportion of “yes” in the
population is given by the slider
value p. (In this example, p =0.75)
Assume that the population
proportion is an unknown, and
we are going to estimate it by
suggesting a 95% confidence
interval based on the data of
one random sample.
Size of this sample is n = 20.
Sample of data
Summary Table
20
0.85
0.079843597
1.0096872
0.69031281
S1 = count
S2 = sp
S3 =
sp
sp
S5 = sp
sp
sp
count
sp
count
pˆ  0.85
The estimated standard deviation of pˆ
is
count
S4 = sp +
The proportion of “yes” in this sample is
pˆ 1  pˆ 
0.851  0.85

 0.080
n
20
sp
Margin of error = 2(0.080) = 0.160
The 95% confidence interval is
(0.85 – 0.16, 0.85 + 0.16) or
(0.69, 1.01)
Based on the result of this sample, we are 95% confident that the
true proportion p lies between 0.69 and 1.01.
0
0.2
0.4
0.6
0.8
1
95% conf. Int.
Note that the interval will vary from sample to sample, but if we
repeat the sampling process indefinitely with samples of the same
size, we will expect 95% of these intervals to capture the true
proportion.
To shorten the interval, we have to increase the sample size.
Note that the interval will vary from sample to sample, but if we
repeat the sampling process indefinitely with samples of the same
size, we will expect 95% of these intervals to capture the true
proportion.
0
0.2
Confidence
Intervals
from
different
samples
0.4
0.6
0.8
1
Use an experiment record sheet to
record more confidence intervals
from other samples of the same
size.
Some intervals may not be
able to capture the true
proportion.
To estimate with a larger
sample, double click on the
‘Sample of Data’ collection
to open its inspector and
adjust the sample size.
Sample of data
Summary Table
80
0.775
The proportion of “yes” in this sample is
0.046687123
0.86837425
0.68162575
S1 = count
S2 = sp
S3 =
sp
sp
is
count
S4 = sp +
S5 = sp
pˆ  0.775 and the sample size is n = 80.
The estimated standard deviation of pˆ
sp
sp
count
sp
count
pˆ 1  pˆ 
0.7751  0.775

 0.0467
n
80
sp
Margin of error = 2(0.0467) = 0.0934
The 95% confidence interval is
(0.775 – 0.0934, 0.775 + 0.0934) or
(0.682, 0.868)
We are now 95% confident that the true proportion lies between
0.682 and 0.868. The interval is shorter when the sample size is
increased from 20 to 80.
Sample size = 80
0
0.2
0.4
0.6
0.8
1
95% conf. Int.
Sample size = 20
0
0.2
0.4
0.6
0.8
95% conf. Int.
1
Example: Halloween Practices and Beliefs
An organization conducted a poll about Halloween practices
and beliefs in 1999. A sample of 1005 adult Americans were
asked whether someone in their family would give out
Halloween treats from the door of their home, and 69%
answered ‘yes’.
Construct a 95% confidence interval for p, the proportion of all
adult Americans who planned to give out Halloween treats from
their home in 1999.
Adapted from Rossman et al. (2001, p.433)
Sample size = 1005
Sample proportion = 0.69
Estimated standard deviation of sample proportions =
pˆ 1  pˆ 
0.69 0.31

 0.0146
n
1005
Margin of error = 2(0.0146) = 0.0292
95% confidence interval is 0.69  0.0292
We are 95% confident that the population proportion lies
between 0.6608 and 0.7192.
0
0.2
0.4
0.6
0.8
95% conf. Int.
1
Example: Personal Goal
According to a survey in a university, 132 out of 200 first-year
students in a random sample have identified “being well-off
financially” as an important personal goal.
Give a 95% confidence interval for the proportion of all firstyear students at the university who would identify being welloff as an important personal goal.
Adapted from Moore & Mccabe (1999, p.597)
Sample size = 200
Sample proportion = 132/200 = 0.66
Estimated standard deviation of sample proportions =
pˆ 1  pˆ 
0.66 0.34

 0.0335
n
200
Margin of error = 2(0.0335) = 0.067
95% confidence interval is 0.66  0.067
We are 95% confident that the population proportion lies
between 0.593 and 0.727.
0
0.2
0.4
0.6
95% conf. Int.
0.8
1
Estimate population mean, with a
confidence interval, from data of a
random sample.

mean  x
mean  x
mean  x
mean  x
Sample
Mean = 
Population
2
n
2
n

There is 95% chance that x
will fall inside the interval
mean  x
2

n
Sample
Mean = 
s.d. = 
Population
ND
2
n
2
n

There is 95% chance that 
will fall inside the interval
mean  x
2
x
n
Sample
Mean = 
s.d. = 
Population
2s
n
2s
n

There is 95% chance that 
will fall inside the interval
mean  x
Sample
2s
x
n
s.d. = s
Mean = 
Population
Open the Fathom file
‘estimate2.ftm’
This summary table record the true
mean and standard deviation of the
population, where are supposed to be
unknowns.
Assume that the population
mean is an unknown, and we
are going to estimate it by
suggesting a 95% confidence
interval based on the data of
one random sample.
Size of this sample is n = 20.
Sample of data
Summary Table
20
29.85
20.228367
4.5232004
38.896401
20.803599
S1 = count
S2 = sm
S3 = ssd
ssd
S4 =
count
S5 = sm +
S6 = sm
The sample mean and standard deviation
are x  29.85 and s  20.23
The estimated standard deviation of x
is
s
20.23

 4.523
n
20
ssd
count
ssd
Margin of error = 2(4.52) = 9.046
count
The 95% confidence interval is
(29.85 – 9.05, 29.85 + 9.05) or
(20.80, 38.90)
Based on the result of this sample, we are 95% confident that the
true mean  lies between 20.80 and 38.90.
0
10
20
30
40
50
95% conf. Int.
Note that the interval will vary from sample to sample, but if we
repeat the sampling process indefinitely with samples of the same
size, we will expect 95% of these intervals to capture the true
mean.
To shorten the interval, we have to increase the sample size.
Example: Protein Intake
A nutritional study produced data on protein intake for women.
In a sample of n = 264 women, the mean of protein intake is
x  59 .6 grams and the standard deviation is s = 30.5 grams.
Estimate the population mean and give a 95% confidence
interval .
Adapted from Bennett et al. (2001, p.401)
Estimated standard deviation of the sample means =
s
30.5

 1.9
n
264
Margin of Error = 2(1.9) = 3.8 grams
95% confidence interval is 59.6  3.8 grams
We can say with 95% confidence that the interval ranging from
55.8 grams to 63.4 grams contains the population mean.
Example: Body Temperature
A study by University of Maryland researchers investigated the
body temperatures of n = 106 subjects. The sample mean of the
data set is x  98.20 F and the standard deviation for the
sample is s  0.62 F .

Estimate the population mean body temperature with a 95%
confidence interval.
Adapted from Bennett et al. (2001, p.403)
Estimated standard deviation of the sample means =
s
0.62

 0.06
n
106
Margin of Error = 2(0.06 F) = 0.12F
95% confidence interval is 98.20 F  0.12F
We can say with 95% confidence that the interval ranging from
98.08F to 98.32F contains the population mean.
Normal Distribution
99.7%
95%
68%
m – 3s
m – 2s
m–s
m
m+s
m + 2s
m + 3s