Question

The pathogen Phytophthora capsici causes bell peppers to wilt and die. Because bell peppers are an important commercial crop, this disease has undergone a great deal of agricultural research. It is thought that too much water aids the spread of the pathogen. Two fields are under study. The first step in the research project is to compare the mean soil water content for the two fields. Units are percent water by volume of soil.

Field A samples, x₁:

10.0	10.5	15.5	10.6	9.9	10.0	16.6
15.1	15.2	13.8	14.1	11.4	11.5	11.0

Field B samples, x₂:

8.1	8.5	8.6	7.3	8.0	7.1	13.9	12.2
13.4	11.3	12.6	12.6	12.7	12.4	11.3	12.5

(i) Use a calculator to calculate x₁, s₁, x₂, and s₂. (Round your answers to two decimal places.)

x1	=
s1	=
x2	=
s2	=

(ii) Assuming the distribution of soil water content in each field is mound-shaped and symmetric, use a 5% level of significance to test the claim that field A has, on average, a higher soil water content than field B.
(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: μ₁ = μ₂; H₁: μ₁ < μ₂H₀: μ₁ > μ₂; H₁: μ₁ = μ₂    H₀: μ₁ = μ₂; H₁: μ₁ > μ₂H₀: μ₁ = μ₂; H₁: μ₁ ≠ μ₂

(b) What sampling distribution will you use? What assumptions are you making?

The standard normal. We assume that both population distributions are approximately normal with known standard deviations.The Student's t. We assume that both population distributions are approximately normal with known standard deviations.    The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.

What is the value of the sample test statistic? (Test the difference μ₁ − μ₂. Do not use rounded values. Round your final answer to three decimal places.)


(c) Find (or estimate) the P-value.

P-value > 0.2500.125 < P-value < 0.250    0.050 < P-value < 0.1250.025 < P-value < 0.0500.005 < P-value < 0.025P-value < 0.005

Answer 1

i) $\bar x1$ = 12.51

   s1 = 2.41

$\bar x2$ = 10.78

s2 = 2.39

ii) a) The level of significance is 0.05.

$H_{0}: \mu1 = \mu2$

$H_{1}: \mu1 > \mu2$

b) The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.

The test statistic is

$t = \frac{\bar x1 - \bar x2}{\sqrt{s1^2/n1 + s2^2/n2}}$

   $= \frac{12.51 - 10.78}{\sqrt{(2.41)^2/14 + (2.39)^2/16}}$

$=1.969$

c)

$DF = \frac{(s1^2/n1 + s2^2/n2)^2}{\frac{(s1^2/n1)^2}{n1 - 1} + \frac{(s2^2/n2)^2}{n2 - 1} }$

         $= \frac{((2.41)^2/14 + (2.39)^2/16)^2}{\frac{((2.41)^2/14)^2}{13} + \frac{((2.39)^2/16)^2}{15} }$

       

P-value = P(T > 1.969)

            = 1 - P(T < 1.969)

            = 1 - 0.9704

            = 0.0296

0.025 < P-value < 0.050