Gaussian Statistics and Confidence Intervals in Population Sampling
Explore Gaussian statistics in population sampling scenarios, understanding Z-based limit testing and confidence intervals. Learn about statistical tests such as F-tests and t-tests through practical examples like fish weight and cholesterol level measurements. Master the calculation of confidence intervals for different levels of certainty using the Normal Distribution model.
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Announcements I [Introduce Self] Pipet Calibration Lab Reports now due 9/20 Quiz 2 On 9/20 (those of you with points under Quiz 2 in SacCT should have been in Lab Procedures Quiz column) Today s Lecture Gaussian Statistics (Chapter 4) Z-based limit testing Confidence Intervals (Z-based and t-based) Statistical Tests (F-test, t-tests) [let me know how far you get on these]
Chapter 4 Gaussian Distributions Now for limit problems example 1 population statistics: A lake is stocked with trout. A biologist is able to randomly sample 42 fish in the lake (and we can assume that 42 fish are enough for proper Z-based statistics). Each fish is weighed and the average and standard deviation of the weight are 2.7 kg and 1.1 kg, respectively. If a fisherman knows that the minimum weight for keeping the fish is 2.0 kg, what percent of the time will he have to throw fish back? (assuming catching is not size-dependent) 1stpart: convert limit (2.0 kg) to normalized (Z) value: Z = (x )/ 2ndpart: use Z area to get percent
Chapter 4 Gaussian Distributions Limit problem example 2 measurement statistics: A man wants to get life insurance. If his measured cholesterol level is over 240 mg/dL (2,400 mg/L), his premium will be 25% higher. His level is measured and found to be 249 mg/dL. His uncle, a biochemist who developed the test, tells him that a typical standard deviation on the measurement is 25 mg/dL. What is the chance that a second measurement (with no crash diet or extra exercise) will result in a value under 240 mg/dL (e.g. beat the test)?
Graphical view of examples Equivalent Area Normal Distribution Table area Desired area 0.45 0.4 0.35 0.3 Frequency 0.25 0.2 0.15 0.1 0.05 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 Z value X-axis 249 240
Chapter 4 Calculation of Confidence Interval 1. 2. Confidence Interval = x + uncertainty Calculation of uncertainty depends on whether is well known When is not well known (covered later) When is well known (not in text) Value + uncertainty = Normal Distribution 0.45 0.4 3. 0.35 0.3 Frequency 4. 0.25 0.2 Z 0.15 x 0.1 n 0.05 Z depends on area or desired probability At Area = 0.45 (90% both sides), 0 -3 -2 -1 0 1 2 3 Z value Z = 1.65 At Area = 0.475 (95% both sides), Z = 1.96 => larger confidence interval
Chapter 4 Calculation of Uncertainty Example: The concentration of NO3- in a sample is measured 2 times and found to give 18.6 and 19.0 ppm. The method is known to have a constant relative standard deviation of 2.0% (from past work). Determine the concentration and 95% confidence interval.
