How to fine the required sample size

In summary, for the second part, you need to use the formula for the variance of the sample mean to solve for the sample size required under the given optimal allocation, taking into account the total sample size calculated from the first part.
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Homework Statement


Given info: cost1=$4, σ1=10, W1=N1/N=0.4
cost2=$9, σ2=20, W2=N2/N=0.6

The second part: Find the sample size required, under this optimal allocation to make
V(ybar) = 1. Ignore the finite population correction factor.

Homework Equations


The equation I used for the first part is nh = n[(Nh)(σh^2)/(ch)]/[(summmation j=1 to H) ((Nj) (σj^2)/(cj))] , where H = 1, 2.

The Attempt at a Solution


The first part of this question asks for the values of n1/n and n2/n that minimize total cost for a given value of the variance, V(ybar). It is a stratified sample.
My answer is n1/n = 0.1818 and n2/n = 0.8181

Can anyone help me with the second part? I'm very lost, any ideas would be appreciated. Thanks
 
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For the second part, you can use the formula for the variance of the sample mean: V(ybar) = (σ^2/n) * (1-W) * (N/N-1), where σ is the population standard deviation and n is the sample size. Since we want V(ybar) = 1, we can rearrange this formula to solve for n: n = (σ^2 * N) / (N-1). Plugging in the given values, we get n = (20^2 * N) / (N-1) = 400N / (N-1). Since we want to minimize cost, we can use the values of n1/n and n2/n from the first part to determine the total sample size, N. The total sample size, N, can be calculated as N = N1/n1 + N2/n2 = (N1 * 0.1818) + (N2 * 0.8181). Once you have N, you can plug it into the formula for n to get the sample size required to make V(ybar) = 1.
 

FAQ: How to fine the required sample size

1. What is the importance of determining the required sample size?

Determining the required sample size is crucial because it ensures that the results of the study are accurate and representative of the entire population. A smaller sample size may lead to biased or unreliable conclusions, while a larger sample size can increase the cost and time of the study.

2. How do I calculate the required sample size?

The required sample size can be calculated using statistical formulas that take into account factors such as the desired level of confidence, margin of error, and population size. There are also online calculators and software programs available to assist with this calculation.

3. What factors should be considered when determining the required sample size?

Some important factors to consider when determining the required sample size include the level of confidence desired, the margin of error allowed, the variability of the population, and the size of the population. Other factors such as the research design and the type of data being collected may also play a role.

4. Can I use a smaller sample size if I have a limited budget or time?

While it may be tempting to use a smaller sample size in order to save time and resources, it is important to ensure that the sample size is large enough to provide reliable results. In some cases, researchers may use techniques such as stratified sampling or power analysis to determine the minimum sample size needed for their study.

5. Is there a standard sample size that should be used for all studies?

No, there is no standard sample size that can be applied to all studies. The required sample size will vary depending on the research question, population, and other factors. It is important for researchers to carefully consider these factors and calculate the appropriate sample size for their specific study.

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