Statistical Concepts(part III)

• Concepts to cover or review today:– Population parameter– Sample statistics– Mean– Standard deviation– Coefficient of variation– Standard error– Confidence interval– Estimating totals– Estimating standard deviation, standard error, confidence intervals,

coefficient of variation for totals– Sample size estimation

• You should understand these terms by the end of the lecture and be able to conduct estimations.

Estimation Problem:• Mice weights (g):

21.219.315.926.1

1816.820.616.210.819.3

1

)(1

2__

n

XXsStdev

n

i

i

n

StdevSE

CI95 = Mean ± 1.96 * SE

n

XXMean

n

i

i 1

__

• Estimate:

Answers

MEAN 18.42STD 4.02

CV 21.8241

se 1.271236

low CI 15.92838up CI 20.91162

Distribution of Means

16 17 18 19 20

Weight (g)

Fre

qu

ency

(n

)

Mean = 18.0SD = 3.6

n = 100

1 se1 se2 se 2 se

2.5 %2.5 %

(truth)

Mean

1 se1 se2 se 2 se1 se1 se2 se 2 se1 se1 se2 se 2 se

Elzinga et al. (2001:76)

Target/Statistical Population

Sample Unit

Individual objects(in this case, plants)

Density to Totals

Estimating Means or Density

1

)(1

2__

n

XXStdev

n

i

i

n

StdevSE

CI95 = Mean ± 1.96 * SE

n

XXMean

n

i

i 1

__

CV = Stdev / Mean

Estimating Totals of Target Population

1

)(1

2__

n

XXNsStdev

n

i

i

totaltotal

n

StdevSE total

total

CI95 = Total ± 1.96 * SEtotal

n

XNXNTotal

n

i

i 1

__

CV = Stdevtotal / Total

CV = Stdevmean / Mean

(both CVs are the same value)

N = Total number of sample units

Example: Total

number of sagebrush

Target/Statistical Population

Sample Unit

Individual objects in sample unit

(sagebrush plants)

Estimating Total Sagebrush Plants

N = 355

n = 25

Estimating Totals

68420

1057920

125

43892068

11473

Mean = 5.40

Stdevmean = 3.48

CV = 0.64

Total = 1917

Stdevtotal = 1230

SEtotal = 246

95% CItotal = 1435, 2399

Sagebrush per sampled plot

Target/Statistical Population

Sample Unit

Individual objects in sample unit

(sagebrush plants)

Want Total to be ± 250 plants our first attempt was ± 482 plants

N = 355

n = 25

• How might we do this ?

Calculating Sample Size

n

StdevSE total

total

CI95 = Total ± 1.96 * SEtotal

total

total

SE

Stdevn

2

total

total

SE

Stdevn

1.96 * SEtotal = 250 SEtotal = 250 / 1.96

Sagebrush Example:

• We want to get within ± 250 shrubs

• How many units do we need to sample ?

2

SE

Stdevn total

SE = 250 / 1.96 = 127.6

Stdevtotal = 1230

n = (1230 / 127.6)2 = 93