1. (a) Let X1,..., Xn be a random sample from a Poisson distribution with parameter λ. Derive a (1 - α)100% large sample con?dence interval for λ.
(b) To date nodes in a phylogenetic tree, the mean path length (MPL) is used in estimating the relative age of a node. The following data represent the MPL for 39 nodes (source: Tom Britton, Bengt Oxelman, Annika Vinnersten, and Kåre Bremer, "Phylogenetic dating with con?dence intervals using mean path-lengths"). Assume that the data (given in centimeters) follow a Poisson distribution with parameter λ.
|
65.2
|
47.0
|
38.2
|
13.5
|
18.0
|
25.6
|
16.3
|
14.0
|
23.2
|
18.8
|
|
7.5
|
13.3
|
11.0
|
54.9
|
22.0
|
50.1
|
32.6
|
26.0
|
13.0
|
9.0
|
|
7.2
|
4.7
|
4.5
|
41.1
|
45.8
|
37.0
|
8.5
|
30.5
|
29.3
|
13.8
|
|
7.7
|
5.5
|
24.1
|
12.5
|
22.3
|
19.0
|
9.5
|
4.7
|
3.0
|
|
Obtain a 95% con?dence interval for λ and interpret.