Question

To determine the organic material in a dried lake ted, the percent carbon by mass is measured at two different locations. In order to compare the means of the two different locations, it must first be determined whether the standard deviations of the two locations are different.

Location1(%C) Location2(%C)
1 10.40 10.10
2 10.20 10.90
3 10.30 10.20
4 10.40 10.70
5 10.20 10.40
a-1. Far each location, calculate the standard deviation and report it.
- location 1: _____
- location 2: _____
a-2. What is the calculated F value for comparing the standard deviation?
b. Are the two standard deviations for the locations significantly different at the 95% confidence level?
A. yes.
B. No.
c. What is the calculated t value used to compare the means of these two locations?
d. Are the means for the two different locations significantly different at the 95% confidence level?
A. yes.
B. No.

Answer 1

a-1.- location 1: =0.01- location 2: =0.113

a-2. F= 0.08849

b. A. yes.

c. t= -1.020

d. B. No.

Explanation:

Location1(%C) Location2(%C)

x x² y y²

1 10.40 108.16 10.10 102.01

2 10.20 104.04 10.90 118.81

3 10.30 106.09 10.20 104.04

4 10.40 108.16 10.70 114.49

5 10.20 104.04 10.40 108.16

∑ 51.5 530.49 52.3 547.51

x`= ∑x/n1 = 51.5/5= 10.3; y`= ∑y/n2= 52.3/5= 10.46

Sx² = 1/n1-1[ ∑x² - (∑x)²/n1] = 1/4[ 530.49 - (51.5)²/5]

= 1/4[ 530.49 - 530.45] = 0.01

Sy² = 1/n2-1[ ∑y² - (∑y)²/n2] = 1/4[ 547.51 - (52.3)²/5]

= 1/4[ 547.51 - 547.058] = 0.113

F= Sx²/Sy²= 0.01/0.113= 0.08849

The critical region is F ≥ F(0.025) (5,5)= 7.15

and F ≤ 1/F(0.025) (5,5) = 1/7.15= 0.13986

Since the calculated value of F =0.08849 falls in the critical region we conclude that the two standard deviations for the locations significantly different at the 95% confidence level.

c. t= x1`-x2`/ sqrt ( s₁²/n₁ + s₂²/n₂)

t= 10.3-10.46/ sqrt( 0.01/5 +0.113/5)

t=- 0.16/0.15684

t= -1.020

d. The calculated value t= -1.020 does not fall in the the critical region

t (0.025) (4) = ±2.776 we conclude that the means for the two different locations are not significantly different at the 95% confidence level.