transcript
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- Chapter 4 Simple Random Sampling n Definition of Simple Random
Sample (SRS) and how to select a SRS n Estimation of population
mean and total; sample size for estimating population mean and
total n Estimation of population proportion; sample size for
estimating population proportion n Comparing estimates
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- Simple Random Samples n Desire the sample to be representative
of the population from which the sample is selected n Each
individual in the population should have an equal chance to be
selected n Is this good enough?
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- Example Select a sample of high school students as follows: 1.
Flip a fair coin 2. If heads, select all female students in the
school as the sample 3. If tails, select all male students in the
school as the sample Each student has an equal chance to be in the
sample Every sample a single gender, not representative Each
individual in the population has an equal chance to be selected. Is
this good enough? NO!!
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- Simple Random Sample n A simple random sample (SRS) of size n
consists of n units from the population chosen in such a way that
every set of n units has an equal chance to be the sample actually
selected.
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- Simple Random Samples (cont.) Suppose a large History class of
500 students has 250 male and 250 female students. To select a
random sample of 250 students from the class, I flip a fair coin
one time. If the coin shows heads, I select the 250 males as my
sample; if the coin shows tails I select the 250 females as my
sample. What is the chance any individual student from the class is
included in the sample? This is a random sample. Is it a simple
random sample? 1/2 NO! Not every possible group of 250 students has
an equal chance to be selected. Every sample consists of only 1
gender hardly representative.
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- Simple Random Samples (cont.) The easiest way to choose an SRS
is with random numbers. Statistical software can generate random
digits (e.g., Excel =random(), ran# button on calculator).
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- Example: simple random sample n Academic dept wishes to
randomly choose a 3-member committee from the 28 members of the
dept 00 Abbott07 Goodwin14 Pillotte21 Theobald 01 Cicirelli08
Haglund15 Raman22 Vader 02 Crane09 Johnson16 Reimann23 Wang 03
Dunsmore10 Keegan17 Rodriguez24 Wieczoreck 04 Engle11 Lechtenbg 18
Rowe25 Williams 05 Fitzpatk12 Martinez19 Sommers26 Wilson 06
Garcia13 Nguyen20 Stone27 Zink
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- Solution Use a random number table; read 2-digit pairs until
you have chosen 3 committee members For example, start in row 121:
71487 09984 29077 14863 61683 47052 62224 51025 Garcia (07)
Theobald (22) Johnson (10) Your calculator generates random
numbers; you can also generate random numbers using Excel
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- Sampling Variability Suppose we had started in line 145? 19687
12633 57857 95806 09931 02150 43163 58636 Our sample would have
been 19 Rowe, 26 Williams, 06 Fitzpatrick
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- Sampling Variability Samples drawn at random generally differ
from one another. Each draw of random numbers selects different
people for our sample. These differences lead to different values
for the variables we measure. We call these sample-to-sample
differences sampling variability. Variability is OK; bias is
bad!!
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- Example: simple random sample n Using Excel tools n Using
statcrunch (NFL)
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- 4.3 Estimation of population mean n Usual estimator
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- 4.3 Estimation of population mean n For a simple random sample
of size n chosen without replacement from a population of size N n
The correction factor takes into account that an estimate based on
a sample of n=10 from a population of N=20 items contains more
information than a sample of n=10 from a population of
N=20,000
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- 4.3 Estimating the variance of the sample mean n Recall the
sample variance
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- 4.3 Estimating the variance of the sample mean
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- 4.3 Example n Population {1, 2, 3, 4}; n = 2, equal weights
SamplePr. of samples2s2 {1, 2}1/61.50.50.125 {1, 3}1/62.0 0.500 {1,
4}1/62.54.51.125 {2, 3}1/62.50.50.125 {2, 4}1/63.02.00.500 {3,
4}1/63.50.50.125 2.2281) =.025 -2.2281.025.95 t 10 P(t <
-2.2281) =.025">
- 0 2.2281 Students t Distribution P(t > 2.2281) =.025
-2.2281.025.95 t 10 P(t < -2.2281) =.025
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- 0 1.96 Standard normal P(z > 1.96) =.025 -1.96.025.95 z P(z
< -1.96) =.025
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- -3-20123 Z t 0123 -2-3 Students t Distribution Figure 11.3,
Page 372
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- -3-20123 Z t1t1 0123 -2-3 Students t Distribution Figure 11.3,
Page 372 Degrees of Freedom
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- -3-20123 Z t1t1 0123 -2-3 t7t7 Students t Distribution Figure
11.3, Page 372 Degrees of Freedom
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- 4.3 Margin of error when estimating the population mean
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- n Understanding confidence intervals; behavior of confidence
intervals.
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- 4.3 Margin of error when estimating the population mean
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- Comparing t and z Critical Values Conf. leveln = 30 z =
1.64590%t = 1.6991 z = 1.9695%t = 2.0452 z = 2.3398%t = 2.4620 z =
2.5899%t = 2.7564
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- 4.4 Determining Sample Size to Estimate
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- Required Sample Size To Estimate a Population Mean If you
desire a C% confidence interval for a population mean with an
accuracy specified by you, how large does the sample size need to
be? n We will denote the accuracy by MOE, which stands for Margin
of Error.
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- Example: Sample Size to Estimate a Population Mean Suppose we
want to estimate the unknown mean height of male students at NC
State with a confidence interval. We want to be 95% confident that
our estimate is within.5 inch of n How large does our sample size
need to be?
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- Confidence Interval for
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- n Good news: we have an equation n Bad news: 1.Need to know s
2.We dont know n so we dont know the degrees of freedom to find t *
n-1
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- A Way Around this Problem: Use the Standard Normal
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- .95 Confidence level Sampling distribution of y
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- Estimating s n Previously collected data or prior knowledge of
the population n If the population is normal or near- normal, then
s can be conservatively estimated by s range 6 n 99.7% of obs.
Within 3 of the mean
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- Example: sample size to estimate mean height of NCSU undergrad.
male students We want to be 95% confident that we are within.5 inch
of so MOE =.5; z*=1.96 n Suppose previous data indicates that s is
about 2 inches. n n= [(1.96)(2)/(.5)] 2 = 61.47 n We should sample
62 male students
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- Example: Sample Size to Estimate a Population Mean - Textbooks
Suppose the financial aid office wants to estimate the mean NCSU
semester textbook cost within MOE=$25 with 98% confidence. How many
students should be sampled? Previous data shows is about $85.
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- Example: Sample Size to Estimate a Population Mean -NFL
footballs n The manufacturer of NFL footballs uses a machine to
inflate new footballs n The mean inflation pressure is 13.5 psi,
but uncontrollable factors cause the pressures of individual
footballs to vary from 13.3 psi to 13.7 psi n After throwing 6
interceptions in a game, Peyton Manning complains that the balls
are not properly inflated. The manufacturer wishes to estimate the
mean inflation pressure to within.025 psi with a 99% confidence
interval. How many footballs should be sampled?
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- Example: Sample Size to Estimate a Population Mean n The
manufacturer wishes to estimate the mean inflation pressure to
within.025 pound with a 99% confidence interval. How may footballs
should be sampled? n 99% confidence z* = 2.58; MOE =.025 n = ?
Inflation pressures range from 13.3 to 13.7 psi n So range =13.7
13.3 =.4; range/6 =.4/6 =.067 12348...
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- Required Sample Size To Estimate a Population Mean n It is
frequently the case that we are sampling without replacement.
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- Required Sample Size To Estimate a Population Mean When
Sampling Without Replacement.
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- 4.3 Estimation of population total
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- n Estimate number of lakes in Minnesota, the Land of 10,000
Lakes.
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- Required Sample Size To Estimate a Population Total
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- 4.5 Estimation of population proportion p n Interested in the
proportion p of a population that has a characteristic of interest.
n Estimate p with a sample proportion. n http://packpoll.com/
http://packpoll.com/
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- 4.5 Estimation of population proportion p
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- Required Sample Size To Estimate a Population Proportion p When
Sampling Without Replacement.
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- 4.6 Comparing Estimates
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- 4.6 Comparing Estimates: Comparing Means
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- 60 Population 1Population 2 Parameters: 1 and 1 2 Parameters: 2
and 2 2 (values are unknown) (values are unknown) Sample size: n 1
Sample size: n 2 Statistics: x 1 and s 1 2 Statistics: x 2 and s 2
2 Estimate 1 2 with x 1 x 2
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- df 0 Sampling distribution model for ? An estimate of the
degrees of freedom is min(n 1 1, n 2 1). Shape?
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- 4.6 Comparing Estimates: Comparing Means
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- 4.6 Comparing Estimates: Comparing Means (Special Case, Seldom
Used)
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- 4.6 Comparing Estimates: Comparing Proportions, Two Cases
Difference between two polls Difference of proportions between 2
independent polls Differences within a single poll question
Comparing proportions for a single poll question, horse-race polls
(dependent proportions)
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- 4.6 Comparing Estimates: Comparing Proportions in Two
Independent Polls
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- 4.6 Comparing Estimates: Comparing Dependent Proportions in a
Single Poll n Multinomial Sampling Situation Typically 3 or more
choices in a poll
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- Worksheet n http://packpoll.com/ http://packpoll.com/
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- End of Chapter 4