What happens if we want several confidence intervals to hold at the same time (concurrently)? Do we still have the same level of confidence we had for each individual interval?(a) Suppose we have two independent random variables x1 and x2 with respective population means μ1 and μ2. Let us say that we use sample data to construct two 80% confidence intervals.Confidence Interval Confidence LevelA1 < μ1 < B1 0.80A2 < μ2 < B2 0.80Now, what is the probability that both intervals hold together? Use methods of Section 5.2 to show thatP(A1 < μ1 < B1 and A2 < μ2 < B2) = 0.64.Hint: We are combining independent events.P(A1 < μ1 < B1 and A2 < μ2 < B2) = P(A1 < μ1 < B1) / P(A2 < μ1 < B2)P(A1 < μ1 < B1 and A2 < μ2 < B2) = P(A1 < μ1 < B1) – P(A2 < μ1 < B2) P(A1 < μ1 < B1 and A2 < μ2 < B2) = P(A1 < μ1 < B1) + P(A2 < μ1 < B2)P(A1 < μ1 < B1 and A2 < μ2 < B2) = P(A1 < μ1 < B1) · P(A2 < μ1 < B2)
What happens if we want several confidence intervals to hold at the same time (concurrently)? Do we still have the same level of confidence we had for each individual interval?
(a) Suppose we have two independent random variables x1 and x2 with respective population means μ1 and μ2. Let us say that we use sample data to construct two 80% confidence intervals.
Confidence Interval Confidence Level
A1 < μ1 < B1 0.80
A2 < μ2 < B2 0.80
Now, what is the
P(A1 < μ1 < B1 and A2 < μ2 < B2) = 0.64.
Hint: We are combining independent events.
P(A1 < μ1 < B1 and A2 < μ2 < B2) = P(A1 < μ1 < B1) / P(A2 < μ1 < B2)
P(A1 < μ1 < B1 and A2 < μ2 < B2) = P(A1 < μ1 < B1) – P(A2 < μ1 < B2)
P(A1 < μ1 < B1 and A2 < μ2 < B2) = P(A1 < μ1 < B1) + P(A2 < μ1 < B2)
P(A1 < μ1 < B1 and A2 < μ2 < B2) = P(A1 < μ1 < B1) · P(A2 < μ1 < B2)
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