Question

statistics and probability questions, solve clearly, show steps and in 30 minutes for thumbs up vote

Given the following sample of measured tree base diameters, for randomly-selected trees from a forest
area (measurements in cm):
41, 39, 46, 44, 15, 55, 39, 42, 56, 38, 64, 34, 59, 31.

a) Calculate confidence intervals for μ = mean base diameter for trees in this forest
area, for:
i. LOC = 95%
ii. LOC = 99%

b) Recalculate the 95% CI, using the same values for x and s but with:
i. n = 4 instead of 14.
ii. n = 100 instead of 14.

c) Estimate the minimum sample size required to generate a 95% CI for μ, with a margin of error within:
i. ±2 cm
ii. ±0.5 cm

Answer 1

i)  LOC = 95%

sample Mean = 43.07
t critical= 2.16
s_M = √(12.68²/14) = 3.39
μ = M ± t(s_M)
μ = 43.07 ± 2.16*3.39
μ = 43.07 ± 7.32

95% CI [35.75, 50.39].

You can be 95% confident that the population mean (μ) falls between 35.7488 and 50.39

ii. LOC = 99%

t critical = 3.01
s_M = √(12.68²/14) = 3.39
μ = M ± t(s_M)
μ = 43.07 ± 3.01*3.39
μ = 43.07 ± 10.20

99% CI [32.87, 53.27].

You can be 99% confident that the population mean (μ) falls between 32.87 and 53.27.

b(i)n=4

t critical = 3.18
s_M = √(12.68²/4) = 6.34

μ = M ± t(s_M)
μ = 43.07 ± 3.18*6.34
μ = 43.07 ± 20.16

95% CI [22.91, 63.23].

You can be 95% confident that the population mean (μ) falls between 22.91 and 63.23.

n=100

t critical = 1.98
s_M = √(12.68²/100) = 1.27
μ = M ± t(s_M)
μ = 43.07 ± 1.98*1.27
μ = 43.07 ± 2.51

95% CI [40.56, 45.58].

You can be 95% confident that the population mean (μ) falls between 40.56 and 45.58.

c )i. ±2 cm

n= (ZC*S/ME)^2

n= (12.68*1.96/2)^2

n= 154.42

n=155
ii. ±0.5 cm

n= (ZC*S/ME)^2

n= (12.68*1.96/0.5)^2

n= 2471