Plants convert carbon dioxide (CO₂) in the atmosphere, along with water and energy from sunlight, into the energy they need for growth and reproduction. Experiments were performed under normal atmospheric air conditions and in air with enriched CO₂ concen- trations to determine the effect on plant growth. The plants were given the same amount of water and light for a four-week period. The following table gives the plant growths in grams: Normal Air: 4.67 4.21 2.18 3.91 4.09 5.24 2.94 4.71 4.04 5.79 3.80 4.38 Enriched Air: 5.04 4.52 6.18 7.01 4.36 1.81 6.22 5.70 On the basis of these data, we want to determine the effect of CO2-enriched atmosphere on plant growth. Let X be the plant growth in enriched air and Y be the plant growth in normal air. Assume X, Y are normal with mean μx, µy respectively, but we don't know their variances. (i) Construct a 95% (two-sided) confidence interval for µx μy by discarding the last four data from the larger sample and then pair up the rest of the two samples in the order they appear. (ii) Using the same method in part (a), construct 95% one-sided confidence intervals for x-μy, i.e. compute & such that P(μx − µy ≤X - Ỹ + ε) = 0.95 and P(ux − uy > X - Y - a) = 0.95.
Plants convert carbon dioxide (CO₂) in the atmosphere, along with water and energy from sunlight, into the energy they need for growth and reproduction. Experiments were performed under normal atmospheric air conditions and in air with enriched CO₂ concen- trations to determine the effect on plant growth. The plants were given the same amount of water and light for a four-week period. The following table gives the plant growths in grams: Normal Air: 4.67 4.21 2.18 3.91 4.09 5.24 2.94 4.71 4.04 5.79 3.80 4.38 Enriched Air: 5.04 4.52 6.18 7.01 4.36 1.81 6.22 5.70 On the basis of these data, we want to determine the effect of CO2-enriched atmosphere on plant growth. Let X be the plant growth in enriched air and Y be the plant growth in normal air. Assume X, Y are normal with mean μx, µy respectively, but we don't know their variances. (i) Construct a 95% (two-sided) confidence interval for µx μy by discarding the last four data from the larger sample and then pair up the rest of the two samples in the order they appear. (ii) Using the same method in part (a), construct 95% one-sided confidence intervals for x-μy, i.e. compute & such that P(μx − µy ≤X - Ỹ + ε) = 0.95 and P(ux − uy > X - Y - a) = 0.95.
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Transcribed Image Text:Plants convert carbon dioxide (CO₂) in the atmosphere, along with water and energy
from sunlight, into the energy they need for growth and reproduction. Experiments were
performed under normal atmospheric air conditions and in air with enriched CO₂ concen-
trations to determine the effect on plant growth. The plants were given the same amount of
water and light for a four-week period. The following table gives the plant growths in grams:
Normal Air: 4.67 4.21 2.18 3.91 4.09 5.24 2.94 4.71
4.04 5.79 3.80 4.38
Enriched Air: 5.04 4.52 6.18 7.01 4.36 1.81 6.22 5.70
On the basis of these data, we want to determine the effect of CO2-enriched atmosphere
on plant growth. Let X be the plant growth in enriched air and Y be the plant growth in
normal air. Assume X, Y are normal with mean x, y respectively, but we don't know
their variances.
µx
(i) Construct a 95% (two-sided) confidence interval for px - y by discarding the last
four data from the larger sample and then pair up the rest of the two samples in the
order they appear.
(ii) Using the same method in part (a), construct 95% one-sided confidence intervals for
µx − µy, i.e. compute ɛ such that
-
P(µx − µy ≤ Ñ − Ỹ + ε) = 0.95
and
P(ux − uy > X - Y - ) = 0.95.
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