Download Sample means - Penn State Department of Statistics

Mar. 29 Statistic for the day:
80.4% of Penn State students
drink; 55.2% engage in “highrisk drinking”
source: Pulse Survey, n = 1446, margin of error = 2.6%
Assignment:
Read Chapter 20
Exercises p265: 1, 2, 3a,b
Sample means: measurement variables
Suppose we want to estimate the mean weight at PSU
Histogram of Weight, with Normal Curve
40
Frequency
30
20
10
0
100
200
300
Weight
Data from stat100.2 survey. Sample size 237.
Mean value is 152.5 pounds.
Standard deviation is about (240 – 100)/4 = 35
What is the uncertainty in the mean?
Suppose we take another sample of 237.
What will the mean be?
Will it be 152.5 again?
Probably not.
Consider what happens if we take 1000 samples
each of size 237 and compute 1000 means.
Histogram of 1000 means with normal
curve, based on samples of size 237
Frequency
100
50
0
145
150
155
Weight
Standard deviation is about
(157 – 148)/4 = 9/4 = 2.25
160
Note: When we have measurement data and we consider
the sample mean, there are two different standard deviations:
1. The original standard deviation of the data. We estimated
that from the original histogram of the data.
2. The standard deviation of the sample mean. We estimated
that from a histogram of 1000 sample means.
In general we will have to be given the standard deviation
of the data. Or we will have to estimate it from a histogram.
But once we have the standard deviation of the data (called
the sample standard deviation) we can skip the histogram of
sample means and use a formula.
Formula for estimating the standard deviation
of the sample mean (don’t need histogram)
Suppose we have the standard deviation of the
original sample. Then the standard deviation
of the sample mean is:
standard deviation of the data
sample size
Jargon: The standard deviation of the mean is also
called the standard error or the standard error of the mean
and abbreviated SEM or SE Mean.
So in our example of weights:
The standard deviation of the sample is about 35.
Write SD = 35.
Sample size is 237
Hence by our formula:
SEM = SD/square root of sample size
Standard error of the mean is 35 divided by
the square root of 237: SEM = 35/15.4 = 2.3
So the margin of error of the sample mean is
2x2.3 = 4.6
Report 152.5 + 4.6 or 147.9 to 157.1
Using the margin of error as 2 SEMs we really have a
95% confidence interval for the pop mean.
Normal Curve of sample mean.
The standard error is 2.3 and the
bell is centered at 152.5.
8
Anatomy of a 95% conf idence interv al
7
6
5
4
3
95% in middle
2
1
2 SEM
0
147.9
152.5
sample mean
157.1
True pop mean in here someplace
The steps for 95% confidence interval:
1. sample mean: 152.5 (given)
2. sample standard deviation: SD = 35 (given)
3. sample size: 237 (given)
4. standard error of the mean: SEM = 35/sqrt(237) = 2.3
(you calculate)
5. number of SEMs for 95% confidence coefficient: 2
(you look up in a normal z table)
Now you put it all together:
6. 95% confidence interval for pop mean: 152.5 + 2x(2.3)
152.5 + 4.6
147.9 to 157.1
Example: Estimate the true population mean amount
spent by stat 100 students for text books in fall 2001.
Include a 98% confidence interval.
From the class sample survey
1. mean: 275 dollars
2. sample standard deviation: SD = 120 dollars
3. sample size: 100
4. standard error of the mean:
SEM = SD/sqrt(100) = 120/10 = 12
5. number of SEMs for 98% confidence interval: 2.33
6. 98% confidence interval: 275 + 2.33x(12)
275 + 27.96
247.04 to 302.96
Interpretation: We estimate that the population of
stat100 students spent about $275.
98% confidence interval is $247 to $303, a reasonable
set of values for the pop mean.
So we believe that the true pop mean amount spent
on books this semester is between $247 and $303 with
our best guess of $275.
Normal Curve of sample mean.
The standard error is $12 and the
bell is centered at $275.
8
7
Anatomy of a 98% confidence
interval
6
5
4
3
98% in
middle
2
1
2.33 SEM
0
$247
$275
sample mean
$303
True pop mean in here someplace
Fibonacci
Guess the next number in the sequence
1, 1, 2, 3, 5, 8,
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …
Called a Fibonacci sequence.
Ratios of pairs after a while equal approximately .618
eg.
8/13 = .615
13/21 = .619
21/34 = .618
34/55 = .618
width
length
If
width
 .618
length
Then the rectangle is called the golden rectangle.
Daisy
Head
21 clockwise spirals
34 counterclockwise
Parthenon in Athens
Villa in Paris by Le Corbusier
St. Jerome
Leonardo
da Vinci
La Parade
Georges Seurat
Place de la Concorde
Piet Mondrian
The golden rectangle has become an aesthetic
standard for western civilization.
It appears in many places:
architecture
art
pyramids
business cards
credit cards
Research question: Do non-western cultures also
incorporate the golden rectangle as an aesthetic standard?
Width to Length ratios for rectangles
appearing on beaded baskets of the Shoshoni
0.662
0.609
0.670
0.600
0.690
0.844
0.606
0.633
0.606
0.654
0.611
0.595
0.570
0.615
0.553
Width to Length ratio of rectangles in Shoshoni
beaded baskets
0.85
0.75
C1
0.693
0.628
0.576
0.610
0.65
Golden Rectangle: .618
0.55
0.749
0.668
0.633
0.652
0.601
0.625
Question: Is the golden rectangle (.618) a reasonable
value for the mean of the population of Shoshoni
rectangles?
1.
2.
3.
4.
sample mean: .638
sample standard deviation: SD = .061
sample size: 25
standard error of the mean: SEM = .012
(I calculated if for you.)
Could you create a 95% confidence interval for the
population mean? (We’d like to know whether
.618 is in this interval.)