1. Suppose we construct a 90%-confidence interval [ΘL, ΘU ] for θ*, that is, P(θ* ∈ [ΘL, ΘU ]) = 90%, which one is/are random? ⃝ Only θ* is random. ⃝ Only ΘL and ΘU are random . ⃝ θ*, ΘL and ΘU are random. Assume that X ∼ N(μ1,σ1), and Y ∼ N(μ2,σ2), where X and Y may be dependent, σ1 and σ2 are unknown. Now, we have paired samples Di = Xi − Yi, i = 1,··· ,n with n = 20. Consider the hypothesis testing, H0 : μ1 = μ2 vsH0 : μ1 ̸= μ2, which test you want to conduct? ⃝ T-test ⃝ Z-test. ⃝ T-test or Z-test. ⃝ need further investigation about the data assumption Let X ∼ N(μ,σ) and X ̄ be sample mean from 25 random samples, then we construct an interval x ̄ ± σ where σ is the population standard deviation of X. Compute the confidence level of this interval estimator, that is, P(X ̄ − σ ≤ μ ≤ X ̄ + σ).
Answer following questions.
1. Suppose we construct a 90%-confidence interval [ΘL, ΘU ] for θ*, that is, P(θ* ∈ [ΘL, ΘU ]) = 90%, which one is/are random?
⃝ Only θ* is random.
⃝ Only ΘL and ΘU are random
. ⃝ θ*, ΘL and ΘU are random.
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Assume that X ∼ N(μ1,σ1), and Y ∼ N(μ2,σ2), where X and Y may be dependent, σ1 and σ2 are unknown. Now, we have paired samples Di = Xi − Yi, i = 1,··· ,n with n = 20. Consider the hypothesis testing, H0 : μ1 = μ2 vsH0 : μ1 ̸= μ2, which test you want to conduct?
⃝ T-test
⃝ Z-test.
⃝ T-test or Z-test.
⃝ need further investigation about the data assumption -
Let X ∼ N(μ,σ) and X ̄ be sample mean from 25 random samples, then we construct an interval x ̄ ± σ where σ is the population standard deviation of X. Compute the confidence level of this
interval estimator , that is, P(X ̄ − σ ≤ μ ≤ X ̄ + σ).
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