題組內容

III. (20%) Let x denote the number of car accidents in a week and the random variable x have a Poisson distribution. We collect a simple random sample of 50 observations and use the sample mean $\bar{x} = \sum x_i / n = 36$ to estimate the population mean of x. Suppose the population standard deviation of x is $\sigma = 6$.

(a) Describe the sampling distribution of \( \bar{x} \). Develop a \( 90\% \) confidence interval for the average number of accidents.