1. 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

2. Today Review Q & A

3. 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

4. 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

5. 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?

6. Example: brook trout length Confidence limits – the concept

7. 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

8. 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

9. Example: brook trout length 4.Tolerance for Type I error α = 10% 5.Write probability statement Verbal: the probability that a line from L 1 to L 2 includes the true mean fork length μ Y of Cat Arm brook trout is equal to 90% Symbolic:

10. Example: brook trout length 6.Plug values into probability statement SE= 1.45 mm α = 10% t 0.05[15] = ? =

11. Example: brook trout length 6.Plug values into probability statement SE= 1.45 mm α = 10% t 0.05[15] = -1.753 7. Make statement about population estimate The limits 51.26 cm to 56.3 cm enclose the true population mean μ Y 90% of the time

12. Confidence limits - comments How do we narrow the confidence interval (i.e. L 2 – L 1 )? 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

13. Review Q & A

14. 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