1. DeMorgan's law states the following: P(A or B)' = P(A' and B')

2. If two events are mutually exclusive then they must be independent.

3. A stratified sample ensures that the sample is representative of all strata.

4. With cluster sampling, the best results occur when elements within strata are internally homogeneous.

5. With stratified sampling, the best survey results occur when elements within strata are internally homogeneous.

So...here is my attempt in terms of tackling these problems.

1. True (for this one I am kind of guessing...I know that for example, NOT (A+B) = A NOT + B NOT, and NOT AB = A NOT + B NOT)

2. FALSE...because of the fact that mutually exclusive means that they cannot occur at the same time, doesn't necessarily mean that they are independent, they can be dependent? Is that a flawed understanding.

3. From what I understand about stratified random sampling, the members of a certain population are divided into two or more homogeneous subsets known as strata, which all share a similar characteristic such as age, gender, etc. A random sample is then selected from each of the strata.

...with this in mind...I would say that the #3 is also TRUE

4. My understanding of cluster sampling is that you are essentially dividing a said population into groups labeled as 'clusters," then randomly select some of these clusters, and include each member of that cluster in the sample.

...I would then think that #4 is FALSE? I would say that they are heterogeneous in nature...but I am not entirely sure.

5. I would say this is TRUE based on the understanding provided in #3

If someone could give me insight as to whether or not I am going in the right direction (if I am wrong and lean me in the right direction that would be great!...thank you!)