Question

Let X₁, ..., Xₙ be i.i.d. with the exponential distribution (parameter λ). To test hypotheses H₀: θ ≥ θ₀, H₁: θ < θ₀, what should be considered?

a) Type I error

b) Type II error

c) Confidence interval

d) Central limit theorem

Answer 1

In hypothesis testing, especially with an exponential distribution, one must consider Type I and Type II errors, confidence intervals, and to a lesser extent, the Central Limit Theorem. Type I error involves wrongly rejecting a true null hypothesis while Type II error involves failing to reject a false null hypothesis.

Step-by-step explanation:

When testing hypotheses, especially with an i.i.d sample X1, ..., Xn from an exponential distribution with parameter λ, it is important to understand different concepts. The question about testing hypotheses H0: θ ≥ θ0, H1: θ < θ0 requires considering several elements:

• Type I error: This occurs if the null hypothesis is true but is incorrectly rejected. It is symbolized by α.
• Type II error: This occurs if the null hypothesis is false but is mistakenly failed to be rejected. It is symbolized by β.
• Confidence interval: Constructing a confidence interval for an estimate can help to understand where the true parameter θ may lie concerning θ0.
• Central limit theorem: Although this might be less directly relevant to hypothesis testing for an exponential distribution, the Central Limit Theorem (CLT) informs us that with a large enough sample size, the sample mean should be approximately normally distributed.

Considering these concepts, one should be careful with the consequences of both types of errors. They are both crucial in hypothesis testing, but their impacts can differ based on the context.