题目

Problem

The continuous random variable D is uniformly distributed over the interval [x – 1, x + 5] where x is a constant.

A random sample of n observations of D is taken, where n is large.

(a) Use the Central Limit Theorem to find an approximate distribution for $\bar D$ Give your answer in terms of n and x where appropriate.

(3)

The n observations of D have a sample mean of 24.6 Given that the lower bound of the 99% confidence interval for x is 22.101 to 3 decimal places,

(b) find the value of n Show your working clearly.

(5)

(Total for Question 7 is 8 marks)

题目中文翻译

连续随机变量 D 在区间 [x – 1, x + 5] 上均匀分布，其中 x 为常数。

从 D 中抽取 n 个观测值的随机样本，且 n 很大。

(a) 利用中心极限定理求 $\bar D$ 的近似分布。如适用，请用 n 和 x 表示你的答案。

这 n 个 D 的观测值的样本平均数为 24.6。已知 x 的 99% 置信区间下限为 22.101，保留到 3 位小数，

(b) 求 n 的值。请清楚写出你的过程。

解答

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先用连续均匀分布的公式求 $D$ 的均值和方差。由于 $n$ 很大，中心极限定理说明样本均值近似服从正态分布，其均值不变、方差除以 $n$ 。

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For the continuous uniform distribution,

\begin{align*} E(D) =&\,\frac{(x-1)+(x+5)}{2}\\ =&\,x+2, \end{align*}

and

\begin{align*} \operatorname{Var}(D) =&\,\frac{\big((x+5)-(x-1)\big)^2}{12}\\ =&\,\frac{6^2}{12}\\ =&\,3. \end{align*}

By the central limit theorem,

\boxed{ \overline D\ \dot\sim\ \operatorname{N}\left(x+2,\frac{3}{n}\right) }.

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样本均值 $24.6$ 估计的是 $x+2$ ，所以 $x$ 的点估计为 $22.6$ 。 $99\%$ 置信区间使用临界值 $2.5758$ ；令“点估计减误差界”等于题给下限 $22.101$ ，即可解出 $n$ 。

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Since $E(D)=x+2$ , the point estimate of $x$ is

\widehat x=24.6-2=22.6.

For a $99\%$ confidence interval, $z=2.5758$ . From part (a), the standard error is

\sqrt{\frac{3}{n}}.

Using the given lower confidence limit,

\begin{align*} 22.6-2.5758\sqrt{\frac{3}{n}} =&\,22.101,\\ 2.5758\sqrt{\frac{3}{n}} =&\,0.499,\\ \sqrt{\frac{3}{n}} =&\,\frac{0.499}{2.5758},\\ n =&\,3\left(\frac{2.5758}{0.499}\right)^2\\ =&\,79.94\ldots. \end{align*}

Since $n$ is a whole-number sample size,

\boxed{n=80}.