Let X represent the number of events that are observed to occur in n units of time or

Chapter 5, Problem 19

(choose chapter or problem)

Let X represent the number of events that are ob- served to occur in n units of time or space, and assume \(X \sim P \text { oisson }(n \lambda)\), where \(\lambda\) is the mean number  of events that occur in one unit of time or space. Assume X is large, so that \(X \sim N(n \lambda, n \lambda)\). Follow steps  (a) through (d) to derive a level \(100(1-\alpha) \%\) confidence interval for \(\lambda\). Then in part (e), you are asked to  apply the result found in part (d).

Show that for a proportion \(1-\alpha\) of all possible samples, \(X-z_{a / 2} \sigma_{x}<n \lambda<X+z_{\alpha / 2} \sigma_{x}\).

Let \(\lambda=X / n\). Show that \(\sigma \lambda=\sigma X / n\)

Conclude that for a proportion \(1-\alpha\) of all possible samples,

\(\lambda-z_{\alpha / 2} \sigma_{\lambda}<\lambda<\lambda+z_{a / 2} \sigma_{\bar{\lambda}}\).

Use the fact that \(\sigma \hat{\lambda} \approx \sqrt{\lambda / n}\) to derive an expression for the level \(100(1-\alpha) \%\) confidence interval for \(\lambda\).

A \(5 m L\) sample of a certain suspension is found to contain 300 particles. Let \(\lambda\) represent the mean number of particles per mL in the suspension. Find a 95% confidence interval for \(\lambda\).

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Text Transcription:

X \sim P \text { oisson }(n \lambda)

\lambda

X \sim N(n \lambda, n \lambda)

100(1-\alpha) \%

\lambda

1-\alpha

X-z_{a / 2} \sigma_{x}<n \lambda<X+z_{\alpha / 2} \sigma_{x}

\lambda=X / n

\sigma \lambda=\sigma X / n

1-\alpha \\

\lambda-z_{\alpha / 2} \sigma_{\lambda}<\lambda<\lambda+z_{a / 2} \sigma_{\bar{\lambda}}

\sigma \hat{\lambda} \approx \sqrt{\lambda / n}

100(1-\alpha) \%

\lambda

5 m L

\lambda

\lambda