Transcript The Diversity of Samples from the Same Population

The Diversity of Samples from the Same Population
Thought Questions
1. 40% of large population disagree with new law. In parts a and b, think about role of sample
size.
a. If randomly sample 10 people, will exactly four (40%) disagree with law? Surprised if only two
in sample disagreed? How about if none disagreed?
b. If randomly sample 1000 people, will exactly 400 (40%) disagree with law? Surprised if only 200
in sample disagreed? How about if none disagreed?
c. Explain how long-run relative-frequency interpretation of probability helped you answer parts a
and b.
2. Mean weight of all women at large university is 135 pounds with a standard deviation of 10
pounds.
a. Recalling Empirical Rule for bell-shaped curves, in what range would you expect 95% of
women’s weights to fall?
b. If randomly sampled 10 women at university, how close do you think their average weight
would be to 135 pounds?
If sampled 1000 women, would you expect average weight to be closer to 135 pounds than for
the sample of only 10 women?
The Diversity of Samples from the Same Population
Setting the Stage
Working Backward from Samples to Populations
• Start with question about population.
• Collect a sample from the population, measure variable.
• Answer question of interest for sample.
• With statistics, determine how close such an answer, based on a sample, would tend to be
from the actual answer for the population.
Understanding Dissimilarity among Samples
We need to understand what kind of differences we should expect to see in various samples
from the same population
The Diversity of Samples from the Same Population
What to Expect of Sample Means
•
Want to estimate average weight loss for all who attend national weight-loss clinic for 10
weeks.
•
Unknown to us, population mean weight loss is 8 pounds and standard deviation is 5 pounds.
•
If weight losses are approximately bell-shaped, 95% of individual weight losses will fall
between –2 (a gain of 2 pounds) and 18 pounds lost.
Possible Samples (random samples of 25 people from this population)
Sample 1: 1,1,2,3,4,4,4,5,6,7,7,7,8,8,9,9,11,11,13,13,14,14,15,16,16
Sample 2: –2, 2,0,0,3,4,4,4,5,5,6,6,8,8,9,9,9,9,9,10,11,12,13,13,16
Sample 3: –4,–4,2,3,4,5,7,8,8,9,9,9,9,9,10,10,11,11,11,12,12,13,14,16,18
Sample 4: –3,–3,–2,0,1,2,2,4,4,5,7,7,9,9,10,10,10,11,11,12,12,14,14,14,19
Results:
Sample 1:
Sample 2:
Sample 3:
Sample 4:
Mean = 8.32 pounds
Mean = 6.76 pounds
Mean = 8.48 pounds
Mean = 7.16 pounds
standard deviation= 4.74 pounds
standard deviation = 4.73 pounds
standard deviation = 5.27 pounds
standard deviation = 5.93 pounds
Each sample gave a different sample mean, but close to 8.
The Diversity of Samples from the Same Population
Conditions for Rule for Sample Means
1. Population of measurements is bell-shaped, and a random sample of any size is measured.
Population mean is equal to 100
The Diversity of Samples from the Same Population
Population Mean = 100. Histograms of sample means from repeated sample of the same size
The Diversity of Samples from the Same Population
2. Population of measurements of interest is not bell-shaped, but a large random sample
is measured. Sample of size 30 is considered “large,” but if there are extreme outliers,
better to have a larger sample.
Population Mean is 21.76
The Diversity of Samples from the Same Population
Population Mean = 21.76. Histograms of sample means from repeated sample of the same size
The Diversity of Samples from the Same Population
If numerous samples or repetitions of the same size are taken, the frequency curve of means
from various samples will be approximately bell-shaped.
The mean for the sampling distribution of the sample mean is equal to the true population mean
The standard deviation(SD) for the sampling distribution of the sample mean is
population standard deviation
sample size
The Diversity of Samples from the Same Population
Normal quantile (probability) plots
One way to assess if a distribution is indeed approximately normal is to plot the data on a
normal quantile plot.
1. Arrange the observed data from smallest to largest. Record what percentile of the data
each value occupies. For example, the smallest observation in a set of 20 is at the 5%
point, the second smallest at the 10% point.
2. Do Normal distribution calculations to find the values of z-scores corresponding to these
same percentiles. For example, z = -1.645 is the 5% point of the standard Normal
distribution, and z = -1.282 is the 10% point.
3. Plot each data point x against the corresponding Normal score. If the data distribution is
close to a Normal distribution, the plotted points will lie close to a straight line.
Systematic deviations from a straight line indicate a non-normal distribution. Outliers
appear as points that are far away from the overall pattern of the plot.
The Diversity of Samples from the Same Population
Normal quantile (probability) plots
Good fit to a straight line: the
distribution of rainwater pH values is
close to normal.
Curved pattern: the data are not normally
distributed. Instead, it shows a right skew: a
few individuals have particularly long
survival times.
The Diversity of Samples from the Same Population
Conditions for Rule for Sample Means
1. Population of measurements is bell-shaped, and a random sample of any size is measured.
OR
2. Population of measurements of interest is not bell-shaped, but a large random sample is
measured. Sample of size 40 is considered “large,” but if there are extreme outliers, better to
have a larger sample.
Text Questions
15. Explain whether you think the Rule for Sample Means applies to each of the following
situations. If it does apply, specify the population of interest and the measurement of interest. If
it does not apply, explain why not.
b. A large corporation would like to know the average income of the spouses of its workers. Rather
than go to the trouble to collect a random sample, they post someone at the exit of the building at
5 P.M. Everyone who leaves between 5 P.M. and 5:30 P.M. is asked to complete a short
questionnaire on the issue; there are 70 responses.
c. A university wants to know the average income of its alumni. Staff members select a random
sample of 200 alumni and mail them a questionnaire. They follow up with a phone call to those
who do not respond within 30 days.
8. Suppose the population of IQ scores in the town or city where you live is bell shaped, with a
mean of 105 and a standard deviation of 15. Describe the frequency curve for possible sample
means that would result from random samples of 100 IQ scores.
The Diversity of Samples from the Same Population
Example : Using Rule for Sample Means
Weight-loss example, population mean and standard deviation were 8 pounds and 5 pounds,
respectively, and we were taking random samples of size 25.
Potential sample means represented by a bell-shaped curve with mean of 8 pounds and
standard deviation:
5
25
= 1 pound
For our sample of 25 people:
• 68% of sample means will be between 7 and 9 pounds
• 95% of sample means will be between 6 and 10 pounds
• 99.7% of sample means will be between 5 and 11 pounds
Increasing the Size of the Sample
Weight-loss example: suppose a sample of 100 people instead of 25 was taken. Potential sample
means still represented by a bell-shaped curve with mean of 8 pounds but standard deviation:
5
100
= 0.5 pounds
For our sample of 100 people:
• 68% of sample means will be between 7.5 and 8.5 pounds
• 95% of sample means will be between 7 and 9 pounds
• 99.7% of sample means will be between 6.5 and 9.5 pounds
The Diversity of Samples from the Same Population
What to Expect of Sample Proportions
A slice of the population:
40% of population carry a
certain gene
Do Not Carry Gene = ,
Do Carry Gene = X
Possible Samples
Sample 1:
Proportion with gene = 12/25 = 0.48 = 48%
Sample 2:
Proportion with gene = 9/25 = 0.36 = 36%
Sample 3:
Proportion with gene = 10/25 = 0.40 = 40%
Sample 4:
Proportion with gene = 7/25 = 0.28 = 28%
The Diversity of Samples from the Same Population
Conditions for Rule for Sample Proportions
1. There exists an actual population with fixed proportion who have a certain trait.
2.
Random sample selected from population (so probability of observing the trait is same for
each sample unit).
3.
Size of sample or number of repetitions is relatively large
(sample size) x (true proportion) and (sample size) x (1 – true proportion) must be greater
that or equal to 5
Defining the Rule for Sample Proportions
If numerous samples or repetitions of the same size are taken, the frequency curve made from
proportions from various samples will be approximately bell-shaped. In other words, the
sampling distribution model of the sample proportion is Normal and centered at the true
population proportion, with standard deviation
(true proportion)(1 – true proportion)
sample size
The Diversity of Samples from the Same Population
Example : Suppose 40% of all voters in U.S. favor candidate X. Pollsters take a sample of 2400
people. What sample proportion would be expected to favor candidate X?
The sample proportion could be anything from a bell-shaped curve with mean 0.40 and standard
deviation:
(0.40)(1 – 0.40) = 0.01
2400
For our sample of 2400 people:
• 68% of sample proportions will be between 39% and 41%
• 95% of sample proportions will be between 38% and 42%
• 99.7% of sample proportions will be between 37% and 43%
Text Question
2. Suppose the truth is that .12 or 12% of students are left-handed, and you take a random sample
of 200 students. Use the Rule for Sample Proportions to draw a picture showing the possible
sample proportions for this situation.
The Diversity of Samples from the Same Population
1000 Simulated Samples (n=30)
S im u la te d D a ta : p = 0.15
200
p  0.15
180
160
n  30
140
120
100
0.15(1  0.15)
80
60
 0.065
30
40
20
Pr o p o r t io n o f Su cce s s e s
0.9333
0.8667
0.8000
0.7333
0.6667
0.6000
0.5333
0.4667
0.4000
0.3333
0.2667
0.2000
0.1333
0.0667
0
0
The Diversity of Samples from the Same Population
1000 Simulated Samples (n=30)
S im u la te d D a ta : p = 0.15
200
180
approximately 95% of sample
proportions fall in this interval.
160
140
120
100
Is it likely we would observe
a sample proportion  0.30?
80
60
40
20
Pr o p o r t io n o f Su cce s s e s
0.9333
0.8667
0.8000
0.7333
0.6667
0.6000
0.5333
0.4667
0.4000
0.3333
0.2667
0.2000
0.1333
0.0667
0
0
The Diversity of Samples from the Same Population
1000 Simulated Samples (n=66)
S im u la te d D a ta : p = 0.15
160
140
120
100
80
60
40
20
Pr o p o r t io n o f Su cce s s e s
0.9 69 7
0.9 09 1
0.8 48 5
0.7 87 9
0.7 27 3
0.6 66 7
0.6 06 1
0.5 45 5
0.4 84 8
0.4 24 2
0.3 63 6
0.3 03 0
0.2 42 4
0.1 81 8
0.1 21 2
0.0 60 6
0
0
The Diversity of Samples from the Same Population
1000 Simulated Samples (n=66)
S im u la te d D a ta : p = 0.15
160
140
approximately 95% of sample
proportions fall in this interval
120
100
(0.062 to 0.238).
80
60
Is it likely we would observe
a sample proportion  0.30?
40
20
Pr o p o r t io n o f Su cce s s e s
0.9 69 7
0.9 09 1
0.8 48 5
0.7 87 9
0.7 27 3
0.6 66 7
0.6 06 1
0.5 45 5
0.4 84 8
0.4 24 2
0.3 63 6
0.3 03 0
0.2 42 4
0.1 81 8
0.1 21 2
0.0 60 6
0
0
The Diversity of Samples from the Same Population
Do Americans Really Vote When They Say They Do?
Reported in Time magazine (Nov 28, 1994):
• Telephone poll of 800 adults (2 days after election) – 56% reported they had voted.
• Committee for Study of American Electorate stated only 39% of American adults had voted.
Could it be the results of poll simply reflected a sample that, by chance, voted with greater
frequency than general population?
Suppose only 39% of American adults voted. We can expect sample proportions to be
represented by a bell-shaped curve with mean 0.39 and standard deviation:
(0.39)(1 – 0.39)
= 0.017 or 1.7%
800
• 68% of sample proportions will be between 37.3% and 40.7%
• 95% of sample proportions will be between 35.6% and 42.4%
• 99.7% of sample proportions will be between 33.9% and 44.1%
The Diversity of Samples from the Same Population
Example: Alzheimer’s in US
The Diversity of Samples from the Same Population
Text Questions
4. A recent Gallup Poll found that of 800 randomly selected drivers surveyed, 70% thought they
were better-than-average drivers. In truth, in the population, only 50% of all drivers can be "better
than average.“
a. Draw a picture of the possible sample proportions that would result from samples of 800
people from a population with a true proportion of .50.
b. Would we be unlikely to see a sample proportion of .70, based on a sample of 800 people, from
a population with a proportion of .50? Explain, using your picture from part a.