Statistical Methods in Natural Resources Management ESRM 304

Transcription

1 Statistical Methods in Natural Resources Management ESRM 304

2 Statistical Methods in Natural Resources Management I. Estimating a Population Mean II. Comparing two Population Means III. Reading Assignment Statistical Methods in Nat. Resources 2

3 I. Estimating a Population Mean A. Simple Random Sampling (SRS) B. Mean, variance, standard dev., std. error C. Estimating Reliability with Confidence Statistical Methods in Nat. Resources 3

4 I. Estimating a Population Mean A. Simple Random Sampling (SRS) 1) The real workhorse of statistical methods Ø All other methods have their roots in SRS 2) Every possible combination of n units is equally probable Ø How to do this? 3) Units may be selected w/ or w/o replacement Statistical Methods in Nat. Resources 4

5 I. Estimating a Population Mean A. Simple Random Sampling - Example Ø A 250-acre forest was sampled for cordwood volume per acre in trees larger than 5 DBH Ø Sample units were chosen to be 1/4-acre, circular plots Ø Sample size was chosen to be n = 25 Ø Sampling without replacement will be used Statistical Methods in Nat. Resources 5

6 I. Estimating a Population Mean B. Mean, variance, standard dev., std. error ü Cordwood volume on the i th plot will be named y i y = 1 y n i = n i= ( ) = = 7 cds per 1 / 4 acre plot Naturally, there are four 1/4-ac plots per acre, so There are 28 cds per acre on average in the forest. Total volume in the forest Ŷ = (cds/acre)(no. of acres)= Ŷ = Ny = 1,000( 7) = 7,000 cds ( ) = 7,000 cds Statistical Methods in Nat. Resources 6

7 I. Estimating a Population Mean B. Mean, variance, standard dev., std. error s y 2 = n i=1 y 2 i 1 n ( n 1) 2 i=1 y i 2 = = ( ) 1 ( ) 2 1,317 1, ( 25 1) = ( cds / qtr.acre) 2 Statistical Methods in Nat. Resources 7

8 I. Estimating a Population Mean B. Mean, variance, standard dev., std. error s y = s y 2 = = cds Standard Error has two variants For sampling with replacement: s y = s y n = = cds For sampling without replacement: s y = s y n 1 n N = ,000 = ( 0.975) = cds Statistical Methods in Nat. Resources 8

9 I. Estimating a Population Mean C. Estimating Reliability with Confidence Ø Ø Ø Ø By itself, the mean does not tell us everything we need to know What remains is knowing how reliable the sample mean is for estimating the population mean Standard Error comes to our assistance here We use a Confidence Interval to tell us the probability of being within a certain range of the population mean Accuracy and Precision interact at this stage Statistical Methods in Nat. Resources 9

10 I. Estimating a Population Mean C. Estimating Reliability with Confidence Ø Ø For large samples, we can appeal to the Normal dist n A 95% CI for the population mean is given by Estimate ± 2( Standard Error of the Estimate) For small samples we rely on Student s t dist n A CI for the population mean is given by ( ) Estimate ± t Standard Error of the Estimate The value of t we choose depends on how confident we wish to be that we have covered the population mean within our range of estimated values (interval) Statistical Methods in Nat. Resources 10

11 I. Estimating a Population Mean C. Estimating Reliability with Confidence Ø The (partial) distribution of t Statistical Methods in Nat. Resources 11

12 I. Estimating a Population Mean C. Estimating Reliability with Confidence Ø For an estimate made from 25 units (24 df), a 95% confidence interval is given by ( ) y ± t s y = 7 ± = 6.20 to 7.80 cds per 1/ 4 ac Expanding that to an acre ( ) 4 7 ± = to cds per acre Statistical Methods in Nat. Resources 12

13 II. Comparing Two Population Means A. Estimating two means, two variances B. Estimating variance and standard error of the difference between the two means C. Two methods to test the difference Statistical Methods in Nat. Resources 13

14 II. Comparing Two Population Means A. Estimating two means, two variances Example - We are interested if our lowland hardwood forest produced the same volume (ft 3 /acre) as our upland hardwood forest over a given time period We sample ten, randomly chosen, one-acre plots in each forest type Statistical Methods in Nat. Resources 14

15 II. Comparing Two Population Means A. Estimating two means, two variances y 1 = The data (y 1 = lowland; y 2 = upland): Lowland Hardwoods Upland Hardwoods Total = 7,370 Total = 3, ,370 ( ) = 737 ft 3 / acre y 2 = 304 ft3 / acre Statistical Methods in Nat. Resources 15

16 II. Comparing Two Population Means A. Estimating two means, two variances s 1 2 = ( ) 1 ( 10 7,370 ) 2 ( 10 1) = 15, s 2 2 = ( ) 1 ( 10 3,040 ) 2 ( 10 1) = 12, Statistical Methods in Nat. Resources 16

17 II. Comparing Two Population Means B. Estimating variance and standard error of the difference between the two means First, estimate a common (pooled) variance s 2 p = s 2 n n 1 1 = ( ) + s 2 ( 2 n 2 1) ( ) + ( n 2 1) ( ) + 12, ( 10 1) ( 10 1) + ( 10 1) 15, = 13, ( ft 3 per acre) 2 Statistical Methods in Nat. Resources 17

18 II. Comparing Two Population Means B. Estimating variance and standard error of the difference between the two means then, the standard error of the difference s y1 y 2 = s p 2 ( n 1 + n ) 2 ( n )( 1 n ) 2 = = ft 3 per acre ( ) ( 10) ( 10) 13, Statistical Methods in Nat. Resources 18

19 II. Comparing Two Population Means C. Two methods to test the difference First, the Confidence Interval method ( ) Estimate ± t Standard Error of the Estimate For a 95% CI (recall we have 18 df) y 1 y 2 ± t s y1 y 2 = ± 2.101( ) = to ft 3 per ac. Statistical Methods in Nat. Resources 19

20 II. Comparing Two Population Means C. Two methods to test the difference First, the Confidence Interval method Estimate ± t Standard Error of the Estimate For a 95% CI ( ) y 1 y 2 ± t s y1 y 2 = ± 2.101( ) = to ft 3 per ac. Statistical Methods in Nat. Resources 20

21 II. Comparing Two Population Means C. Two methods to test the difference Second, the t test (Hypothesis Testing) method The hypothesis we actually test states there is no difference between the means (the null ) To run the test, we choose a probability of incorrectly rejecting the notion of no difference let s say 5% t obs = y 1 y 2 s y1 y 2 = = Is this more extreme than a 5% t with 18 df? YES! Statistical Methods in Nat. Resources 21

22 III. Reading Assignment A. Elementary Statistical Methods for Foresters, by Freese, p. 1-13; B. READ This! Study Tips, by Rose Statistical Methods in Nat. Resources 22