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BA 253: Business Statistics
Today
Sampling Distributions
Confidence Intervals
9/30/15
Fri
More CI
ICE 5
↑ Descriptive Statistics: Summarize Data ↑
↓ Inferential Statistics: Estimate Population from Sample ↓
Chapter 7: Sampling and Sampling Distributions
Ex:
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Collect n data, calculate sample mean = x = point estimate.
Use simple random sample.
Is x ≈ μ??? That is the question!
The approximation gets better as the sample size, n, increases.
How is x distributed? In other words, what is the probability distribution of the
sample mean? Guess = _________________
Central Limit Theorem
For large sample sizes (typically, n ≥ 30), the sample mean x is approximately normally
distributed.
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CLT is true regardless of the original distribution of the data.
As n↑, the standard deviation decreases. (That is, x gets closer to μ.)
This is why the normal distribution is so common and so important.
Sampling Distribution of x
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The sample mean, x , has mean μ, standard deviation
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and is
n
o Normally distributed if the underlying distribution is Normal.
o Approximately Normally Distributed whatever the underlying
distribution is if n ≥ 30.
o As the sample size increases, n ↑, the sample mean gets closer and closer
to the population mean, x ≈ μ.
Show Galton board video and Plinko video…
Ex: Normal with mean 100 and sd 20.
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How close is your result to 100? Is one data point enough? No.
Get 5 numbers, then average. Closer.
On Excel, get 30, 100, 1000 numbers and then average.
As sample size increases, the approximation gets better.
Confidence Intervals for Mean
Ex: How much do FLC students spend on gas per week?
Survey 50 students and get: x = $37, s = $19. Is μ ≈ x = $37?
How close is the sample mean to the population mean?
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With 95% certainty, what is the population mean?
o Show/discuss formula.
o (Despite having only surveyed 50 people) we are 95% certain that the
population mean µ (the average for the entire population) is somewhere in
the range from [$32, $42].
Want to reduce the size of the interval? Collect more data!
Common Confidence Levels:
Confidence
90%
95%
98%
99%
z-score
± 1.645
± 1.96
± 2.33
± 2.575
Show where these come from.
Sample Size Determination:
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What if we want the error to only be ± $2.5? How many samples n do we need?
o E = ………. n = ………. = 222
o What if we want ± $1??? n = 1387.
Error
± $5
± $2.5
± $1
± $0.1
Sample Size
n = 50
n = 222
n = 1387
n = 138682
Ex: How many texts do college students send daily?
 Collect data.
 n < 30, but assume Normal anyway, probably reasonable.
 Determine a 95% confidence interval.
 What sample size would be needed to know within ± 10???
Confidence Interval for Proportions:
Ex: Same survey as before. With higher gas prices, have you considered switching to a
more fuel efficient vehicle? p = 37/50 = 74% = 0.74
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Is p = p = 74%?
With 95% certainty, what is p?
 Point Estimate: p = 74%.
 Confidence Interval: p ± E = …… = 74% ± 12% = (61%, 85%)
How many data necessary to know p within ± 5%?
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 E = ……, solve for n, = 303
What if we didn’t know p ? Then assume p = ½.
Sample Size Determination:
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At 95% confidence, how many samples n necessary to know a proportion within
± 5%? ± 1%?
Quick estimate
Fill in………..
Point Estimate
Confidence Interval
How many samples?
Mean
Proportion