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Apr. 8 Statistic for the day:
In the United States in 2001...
Units of blood donated: 15 million
Number of donors: 8 million
Units transfused: 14 million
Number of patients: 4.9 million
Assignment: Prepare for midterm
Speaking of college basketball...
For each of the following, say whether you are a
fan of that sport or not: College basketball
41% answered yes or somewhat.
The fine print (from gallup.com):
- 3% margin of error; sample size=
In a Gallup poll conducted Dec. 5-8, 2004, people
were asked:
Formula for estimating the standard
deviation of a sample proportion (don’t
need histogram):
sample proportion (1 sample proportion )
sample size
× −
× −
If we happen to know the true population proportion we use it instead of the sample proportion.
What to expect from sample proportions
Facts: fingerprints may be influenced by prenatal hormones.
Most people have more ridges on right hand than left.
People who have more on the left hand are said to have leftward asymmetry.
Women are more likely to have this trait than men.
The proportion of all men who have this trait is about 15%
In a study of 186 heterosexual and 66 homosexual men 26 (14%) heterosexual men showed the trait and 20 (30%) homosexual men showed the trait
(Reference: Hall, J. A. Y. and Kimura, D. "Dermatoglyphic Asymmetry and Sexual Orientation in Men", Behavioral Neuroscience, Vol. 108, No. 6, 1203-1206, Dec 94. )
Is it unusual to observe a sample of 66 men and observe a sample proportion of 30%?
We now know what the distribution of sample
proportions based on a sample of 66 should look like.
We will suppose that the true proportion in the
population of men is 15%.
Standard × −
deviation
Thus, a sample proportion of 30% would be
(.30-.15)/.044 = 3.41 standard deviations above
the true mean, assuming that the sample is a
representative sample from the population.
0.0 0.1 0.2 0.
0
5
10
15
Frequency
Histogram of proportions, with Normal Curve n = 66, true proportion = .15, standard deviation =.
homosexual men 0.062 0.15 0. 4 standard deviations
2 std devs
The sample proportion for homosexual men (30%) is too large to come from the expected distribution of sample proportions.
Sample means: measurement variables
Data from stat 100 survey, spring 2004. Sample size 237. Mean value is 152.5 pounds. Standard deviation is about (240 – 100)/4 = 35
Suppose we want to estimate the mean weight at PSU
100 200 300
40
30
20
10
0 Weight
Frequency
Histogram of Weight, with Normal Curve
Standard deviation is about (157 – 148)/4 = 9/4 = 2.
145 150 155 160
100
50
0 Weight
Frequency
curve, based on samples of size 237
Histogram of 1000 means with normal
Hypothetical result, using a “population” that resembles our sample:
Extremely
interesting:
The histogram
of means is
bell-shaped,
even though
the original
population
was skewed!
Formula for estimating the standard deviation of
the sample mean (don’t need histogram)
Just like in the case of proportions, we would like to have a simple formula to find the standard deviation of the mean without having to resample a lot of times.
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
Example: SAT math scores
Suppose nationally we know that the SAT math test has a mean of 100 points and a standard deviation of 100 points.
Draw by hand a picture of what you expect the distribution of sample means based on samples of size 100 to look like.
Sample means have a normal distribution mean 500 standard deviation 100/10 = 10
So draw a bell shaped curve, centered at 500, with 95% of the bell between 500 – 20 = 480 and 500 + 20 = 520
A sample of 100 SAT math scores with a mean of 540 would be very unusual.
A sample of 100 with a mean of 510 would not be unusual.
460 480 500 520 540
Normal curve of SAT means, sample size 100
Score
Salk observed 42 rhesus monkeys in Bronx Zoo holding babies. 40 held the baby on left.
Suppose this is a sample of Rhesus monkeys. Find a 90% confidence interval for the population proportion of monkey mothers who hold baby on left.
- sample proportion: 40/42 =.
- sample size: 42
- standard deviation of sample proportion:.
- number of standard deviations for 90%: 1.
- 90% confidence interval: .95 ± 1.64×(.034) .95 ±. .895 to 1. .895 to 1
Study 1: monkeys (^) Study 2: mothers both right and left handed
Of 255 right handed mothers, 83% held baby on left. They said it was more natural since it frees the right hand for doing things.
Of 32 left handed mothers, 78% held baby on left. They said it was better to hold baby in dominant arm.
Right handed: 98% confidence interval
- sample proportion:.
- sample size: 255
- standard deviation of the sample proportion:.
- number of standard deviations for 98%: 2.
- 98% confidence interval: .83 ± 2.33×(.024) .83 ±. .774 to.
Left handed mothers 90% confidence interval:
- sample proportion:.
- sample size: 32
- standard deviation:.
- number of standard deviations for 90%: 1.
- 90% confidence interval: .78 ± 1.64×(.073) .78 ±. .66 to.
Study 3: shoppers
Researchers loitered around a supermarket parking lot and recorded in which arm the shoppers carried their grocery bags.
Of 438 shoppers, 50% carried bags on left.
95% confidence interval:
- sample proportion:.
- sample size: 483
- standard deviation of sample proportion:.
- number of standard deviations for 95%: 2
- 95% confidence interval: .50 ± 2×(.024) .50 ±. .452 to.
Study 4: paintings and sculpture
Of 466 paintings and sculpture of the Madonna and child, 80% held baby on left.
98% confidence interval:
- sample proportion:.
- sample size: 466
- standard deviation of sample proportion:.
- number of standard deviation for 98%: 2.
- 98% confidence interval: .80 ± 2.33×(.019) .80 ±. .756 to.
art lf t handedmonkey srt handed shoppers
Conf idence interv als f or proportion holding item on lef t
n=466 n=32 n=42 n=255 n=
98% 98% 90%
90%
95%
What makes for wider intervals? Smaller samples, larger confidence coefficients
Summary
Example: Estimate mean # of pairs of
jeans owned by a student at PSU Histogram of Jeans
Jeans
Frequency
0 10 20 30 40
0
10
20
30
40
50 Mean = 7.8 pairs St. Dev. = 5.8 pairs
Sample size = 222
Give a 98% confidence interval.
Example: Estimate mean # of pairs of
jeans owned by a student at PSU
Mean = 7.8 pairs
St. Dev. = 5.8 pairs
Sample size = 222
Give a 98% confidence interval.
SEM 0. 222
= =
of SEMs for 98% confidence: 2.
98% confidence interval:
7.8 ± 2.33 ×0.
7.8 ± 0.9, or 6.9 to 8.
Interpretation: We estimate that the population of Penn State students owns 7.8 pairs of jeans on average.
98% confidence interval is 6.9 to 8.7 pairs, a reasonable range of values for the true (population) mean.
Guess the next numbers in the sequence
1, 1, 2, 3, 5, 8, 13,
Called a Fibonacci sequence.
Ratios of pairs after a while equal approximately.
eg. 8/13 =. 13/21 =. 21/34 =.
Fibonacci Sequence
21, 34, ...
width
length
=. 618 length
width If
then the rectangle is called a golden rectangle.
Width to Length ratios for rectangles appearing on beaded baskets of the Shoshoni
0.693 0.662 0.690 0.606 0.570 0.749 0.652 0. 0.609 0.844 0.654 0.615 0.668 0.601 0.576 0. 0.606 0.611 0.553 0.633 0.625 0.610 0.600 0.
C
beaded baskets
Width to Length ratio of rectangles in Shoshoni
Golden Rectangle:.
SE of diffSE of diff
SEMSEM
SDSD
samplesample sizesize
meansmeans
musiciansmusicians nono perfperf pitchpitch
musiciansmusicians perfperf pitchpitch
Diff in means = -.57 – (-.23) = -. 95% CI: -.34 ± 2 ×.043, which is -.43 to -. Conclusion: They are not close. There is a difference.
2 2 .019 + .039 =.
General conclusions:
There is a significant difference between the asymmetry of the PT for musicians with perfect pitch and both musicians without perfect pitch and non-musicians.
This strongly suggests that there is a relationship between the physical structure of the PT in the brain and perfect pitch ability.
Categorical ordinal, categorical nominal, quantitative discrete, or quantitative continuous?
Eye colorEye color WeightWeight ^ Number of siblingsNumber of siblings GenderGender ^ Time in 100Time in 100--meter dashmeter dash Number of cigarettes smoked in a dayNumber of cigarettes smoked in a day ^ Building where your first class occursBuilding where your first class occurs Year in school (Year in school (frfr / so // so / jrjr // srsr))
(CN)(CN)
(QC)(QC)
(QD)(QD)
(CN)(CN)
(QC)(QC)
(QD)(QD)
(CN)(CN)
(CO)(CO)
Consider a clock that’s 5 minutes fast.
Valid or invalid?Valid or invalid? Reliable or unreliable?Reliable or unreliable? (^) Biased or unbiased?Biased or unbiased?
Answer: valid, reliable and biased.
Consider a scale that is sometimes
several pounds too low, sometimes
several pounds too high
^ Valid or invalid?Valid or invalid? Reliable or unreliable?Reliable or unreliable? Biased or unbiased?Biased or unbiased?
Answer: valid, unreliable and unbiased.
