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• Today: Lab 3 & A3 due
• Mon Oct 1: Exam I  this room, 12 pm
Please, no computers or smartphones
• Mon Oct 1: No grad seminar
Next grad seminar: Wednesday, Oct 10
• Next Lab: Tuesday Oct 2
Today
Review Q & A
Confidence limits
Used to evaluate the uncertainty of an estimate
made from data
A confidence interval gives an estimated range of values
which is likely to include an unknown population
parameter
Example: brook trout length
Cat Arm Lake (Great Northern Peninsula) was flooded to
create a reservoir for a Hydroelectric Generating Station
Potential impact of flooding: reduction in recruitment of fish
If that happened, NL Hydro would build hatchery = $$$
Measure size of 0-group to establish baseline for comparison
after flooding
Example: brook trout length
Quantity:
Fork length Y = mm
n=16
Total population of 0-group: ca 700
Sampling fraction = 16/700 ≈ 2%
Sample mean
mean(Y)=53.8 mm
Estimate of true mean E(Y)  unknown
How reliable is our estimate of the mean?
Example: brook trout length
Confidence limits – the concept
Example: brook trout length
Table 7.5a Generic recipe for calculating a confidence limit
1. State population, state statistic
2. Calculate statistic from data
3. Determine distribution of the estimate
4. State tolerance for Type I error
5. Write a probability statement about the estimate
6. Plug values into statement to obtain confidence limits
7. Make a statement that the limits include the true value of
population parameter
Example: brook trout length
1. Population: all brook trout < 1 year in Cat Arm Lake in 1982
Statistic: population mean length
2. Calculate statistic
mean(Y)=Y=53.8 mm
3. Distribution of estimate
Key  Table 7.2
Statistic is population mean
data cluster around central value
sample size is small (n<30) ……..t distribution
Example: brook trout length
4. Tolerance for Type I error
α = 10%
5. Write probability statement
Verbal: the probability that a line from L1 to L2 includes the true
mean fork length μY of Cat Arm brook trout is equal to 90%
Symbolic:
PL1  Y  L2   1  
P  Y  t( / 2)[ n1] SE  Y  Y  t(1 / 2)[ n1] SE 1  
SE 
s
n
Example: brook trout length
6. Plug values into probability statement
P  Y  t( / 2)[ n1] SE  Y  Y  t(1 / 2)[ n1] SE 1  
Y  53.8mm
SE= 1.45 mm
α = 10%
t0.05[15] = ? =
Example: brook trout length
6. Plug values into probability statement
Y  53.8mm
SE= 1.45 mm
α = 10%
t0.05[15] = -1.753
P  53.8  1.753 *1.45  Y  53.8  1.753 *1.45  90%
P  51.26  Y  56.34  90%
7. Make statement about population estimate
The limits 51.26 cm to 56.3 cm enclose the true population
mean μY 90% of the time
Confidence limits - comments
How do we narrow the confidence interval (i.e. L2 – L1)?
1. increase α
2. increase n
3. decrease σ
For many statistics the distribution of the estimate is
unknown
Solution: generate an empirical distribution by
resampling  bootstrap
Review Q & A
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Quantity
Measurement scale
Dimensions & Units
Equations
Data Equations
– Sums of squared residuals quantify improvement
in fit, compare models
• Quantify uncertainty through frequency distributions
– Empirical
– Theoretical
– 4 forms, 4 uses
• Hypothesis testing
• Confidence interval