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QuestionStatistics
"Suppose T and Z are random variables.
a. If P(T>2.81)= 0.08 and P(T<−2.81)= 0.08, obtain P(−2.81≤T≤2.81).
b. If P(−0.84≤Z≤0.84)= 0.60 and also P(Z>0.84)=P(Z<−0.84), find P(Z>0.84)."
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Answer
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Step 1:I'll solve this probability problem step by step, following the specified LaTeX formatting guidelines:
Part a:
Step 2:: Understand the given probabilities
- $$P(T > 2.81) = 0.08
- P(T < - 2.81) = 0.08
Step 3:: Recognize the symmetry of the probabilities
- The probabilities are symmetric around zero - This suggests T follows a symmetric distribution
Step 4:: Calculate the probability of the central region
- $$P(-2.81 \leq T \leq 2.81) = 0.84
- P(- 2.81 \leq T \leq 2.81) = 1 - 0.08 - 0.08 Part b:
Step 5:: Understand the given probabilities
- $$P(-0.84 \leq Z \leq 0.84) = 0.60
- P(Z > 0.84) = P(Z < - 0.84)
Step 6:: Use the symmetry condition
- Let $$x = P(Z > 0.84)
- Then P(Z < - 0.84) = x
Step 7:: Calculate the total probability
- $$0.60 + 2x = 1
- 0.60 + x + x = 1
Step 8:: Solve for x
- $$x = 0.20
- 2x = 0.40
Final Answer
a. P(- 2.81 \leq T \leq 2.81) = 0.84 b. P(Z > 0.84) = 0.20
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