I’m recording this bug and asking if anyone else has encountered something similar.
In the Thinking through confidence intervals step, the thought is in Korean:
Thought for 45 seconds
Assessing interval behavior
I’m thinking through the concept of a confidence interval for λ and recognizing that the interval I is for 1/λ, not λ.
Examining confidence interval
Hm, that's interesting. I'm looking at the conservative confidence interval for an exponential model, determining it as a range for λ, considering the sample size and observed mean.
Thinking through confidence intervals
Te 정보를 보니, λ̂n는 λ의 일관적이고 비대칭적으로 접근하는 추정치 같아. That's 일치하는 λ CI의 중심을 다루는 것인 것 같은데.
Clarifying the interval
I’m piecing together why the interval I isn't a confidence interval for λ. It seems to either focus on 1/λ, hinge on the true parameter, is random but constructed otherwise, or could be invalid if endpoints reverse.
Understanding confidence intervals
I'm thinking about why the interval I is not a confidence interval for λ. It turns out the interval's endpoints depend on the true parameter λ, which isn't known, making it invalid for precise data analysis.
Evaluating interval validity
I’m considering the validity of a confidence interval. Since endpoints hinge on the true parameter and data alone can't determine them, the interval must align with the true parameter.
Finding the interval
I’m calculating a conservative confidence interval for λ using q_{α/2} and n. It broadens as λ stretches to infinity, leading to options like the empty interval or a point estimate.
Charting confidence intervals
I’m calculating explicit confidence intervals (CIs) for an exponential statistical model, focusing on λ. Truncation and narrowing intervals are key aspects here.
Determining critical value
I calculated the critical value for q_{α/2} at a 90% confidence level by looking up the standard normal quantile and truncating it to two decimal places.
Analyzing variance
I’m working through the asymptotic distribution of λ̂n, noting it converges to N(λ, λ^2 / n). The task is to manage variance in bounds involving λ. Hm, that's interesting, aligning bounds or utilizing confidence intervals might be key.
Determining λ̂n
I started by identifying the standard error method to substitute λ̂n for λ, and subsequently, I solved the inequality |λ̂n - λ| ≤ q_{α/2} λ / √n. Calculating λ̂n using the given sample mean (7.8), I truncated to four decimal places, resulting in λ̂n = 0.1282.
Mapping the problem
Let's break down an inequality to find λ. Simplify and solve for λ by isolating it and using algebraic steps.
Rewriting and simplifying
Let's see, I’m rearranging the inequality by isolating terms with λ on one side. This leads me to a tighter expression for λ̂n - λ.
Shaping the inequality
I’m working through transforming the inequality to bound λ using δ and λ̂n. This method helps establish the constraints on λ.
Calculating confidence intervals
I began by computing δ and λ̂n. From there, I determined the lower and upper bounds (λ_lower and λ_upper) using the 'solve method' and truncated results to four decimal places.
Calculating confidence intervals
First, I calculate the standard error, then the margin of error. Using the plug-in method, I determine the confidence interval [0.1020, 0.1544] and compare it to the solve method confidence interval.
Pinpointing the narrower confidence interval
I calculated the widths: the 'plug-in method' interval is 0.0524, while the 'solve method' is 0.0549. The 'plug-in method' interval is narrower and centered around λ̂n.