Question

A state forest service is conducting a study of the people who use state-operated camping facilities. The state has two camping areas, one located in the mountains and one located along the coast. The forest service wishes to estimate the average number of people per campsite during a popular weekend when all sites are expected to be used. There are 50 campsites in the mountains and 100 along the coast. From experience, the forest service knows that most sites in the mountains contain from 1-6 people and most sites along the coast contain from 1-10 people. Suppose the forest service can afford to conduct 20 interviews (e.g. they can select 20 campsites) (because they are asking many other questions in addition to number of people using the campsite and they want to do the interviews in person).
Using Neyman allocation, how would you suggest they allocate the 20 interviews among these 2 strata to minimize the standard error of the estimate of the overall average number of people per campsite?

Answer 1

Answer:

Allocation: 4 samples should be from the mountain and 16 from along the coast.

Step-by-step explanation:

Neyman allocation is technique of sample allocation used in cases of stratified sampling.

The formula to compute the best sample size of each stratum is:

$n_{h}=n\times \frac{(N_{h}\times SD_{h})}{\sum\limits^{k}_{i=1}(N_{i}\times SD_{i})}$

The information provided is:

$N_{m}=50\\N_{c}=100\\n=20$

Compute the range for the number of people at the mountain campsite as follows:

$R_{m}=6-1=5$

Then the standard deviation for the number of people at the mountain campsite will be:

$SD_{m}=\frac{R_{m}}{4}=\frac{5}{4}$

Compute the range for the number of people along the coast campsite as follows:

$R_{c}=10-1=9$

Then the standard deviation for the number of people along the coast campsite will be:

$SD_{c}=\frac{R_{c}}{4}=\frac{9}{4}$

Compute the sample size for the mountain campsite as follows:

$n_{m}=n\times \frac{(N_{m}\times SD_{m})}{(N_{m}\times SD_{m})+(N_{c}\times SD_{c})}$

    $=20\times \frac{(50\times (5/4))}{(50\times (5/4))+(100\times (9/4))}\\\\=20\times 0.2174\\\\=4.348\\\\\approx 4$

Compute the sample size for along the coast campsite as follows:

$n_{c}=n-n_{m}=20-4=16$

Thus, 4 samples should be from the mountain and 16 from along the coast.