Transcript Confidence Intervals: Estimating a Population Mean, μ

Confidence Intervals:
Estimating a Population Mean, μ
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Want to estimate average age of San Diego college students
Don’t want to sample EVERY STUDENT
Instead, I could take a sample (e.g., this class) & estimate μ
Xbar! This is called a “point estimate”
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The purpose of a CI is to give an “interval estimate” of
the population mean, μ with some degree of certainty
e.g., “We are 95% confident that the mean age of all
San Diego college students is between 18-24 years old”
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Confidence Intervals
Used when we can only collect a sample from a large
population
If we had data from the entire population, we could say
EXACTLY what the mean value is with 100% confidence
A Confidence Interval (CI) has two parts:
Size: range of number values (i.e., 7.3-8.4)
Confidence Level (i.e., 95%)
Trade-off between confidence and size of confidence
interval
If I want to be 100% confident in my estimate, would my
range be very small or very large?
If I only need to be 10% confident, my range can be small
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What do CI’s Mean?
CI  X  z * ( X )
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What confidence intervals DO NOT mean:
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“There is a 95% chance that my CI includes the true
population mean” - NOT TRUE!
Either it DOES or it DOESN’T!
“I am 95% confident that my CI includes the
true population mean”
Even better: “If I ran this study 100 times, and
made 100 CI’s in this manner, 95 of my 100
CI’s would include the true population mean”
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Confidence Interval
Example
z test example: Does the mean GPA of
students who take statistics differ from the
mean GPA of all college students? Construct
a 95% confidence interval, given: μ = 2.7, σ =
0.6, Xbar = 2.85, sample size n = 36.
σXbar = 0.6/√36 = 0.1
μ = 2.7
z* = -1.96
z* = +1.96
Confidence Intervals
CI  X  z * ( X )
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Xbar = 2.85
z = ±1.96
σXbar = 0.6/√36 = 0.1
–Xbar = 2.85 ± 1.96*(0.1)
–Xbar = 2.85 ± 0.196
–Xbar = between 2.654 and 3.046
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So What???
“We are 95% confident that the population mean lies
between 2.654 and 3.046.”
The confidence interval we constructed based on our
sample includes the overall population mean: 2.70
This is another way of saying that this sample and
our population mean aren’t significantly different!
That is, we shouldn’t reject H0
2.70
2.654
2.85
σXbar = 0.1
3.046
Confidence Interval Formulas
z test CI:
CI  X  z * ( X )
Single Sample t CI:
CI  X  t * ( s X )
Dependent
 t CI:
CI  D  t * ( s D )
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Independent t CI:
CI  ( X 1  X 2 )  t * (s X
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X 2
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Ch. 12 hw
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A sample of 25 statistics students eats a mean of 9 candy bars per month; it is
known that the population mean for college students is 8, with a standard
deviation of 4.
1a. Construct a 95% confidence interval for the true population mean of statistics
students’ candy bar eating.
b. Does this CI indicate a significant difference between statistics students and
other students (why or why not)?
c. Compare & contrast this analysis with a 2-tailed z-test on the same data.
2. Say a different sample from another college yielded a 95% confidence interval of
8.5 ± 1.5. True or false:
a. The true population mean is between 7 and 10.
b. About 95% of these students sampled scored between 7 and 10.
c. There is a 95% chance that this CI contains the true population mean.
d. 5% of confidence intervals constructed this way will not contain the true
population mean.
3. The college candy salesman wants a more precise confidence interval to better
estimate inventory. Name two ways to obtain a more precise CI.