Which of the following are required for an independent random sample? Check all that apply.

a. Every time an individual is selected for the sample, each of the remaining individuals has a lower chance of being included.
b. Sampling must be done with replacement.
c. Every individual in the population has the same probability of being selected for the sample.
d. Every time an individual is selected for the sample, each of the remaining individuals has a higher chance of being included.

Choice b and choice c are required for an independent random sample.

explanation:

Sample selected randomly from a population is random sample. Independent random samples are samples, collected from two different datas or samples. If one sample has been selected , it has no influence on the selection of another sample.

Even if one or more samples are selected , its probability remains constant for all selections. Independent random sample allows for replacement also.

The disadvantage of such a sample process is, that it can lead often to sampling errors.

The advantage of such sampling method, is that it helps reduce the cost and helps in including important subpopulations.

i'm assuming that it's c.)   volcanoes built the mountains and earthquakes tore them down.

let x and 400 - x be the two numbers.  

==> 0.80x + 0.85(400 - x) = 400 - 68 from the statement of the problem.  

< ==> 0.80x + 0.85(400 - x) = 400 - 68  

==> 340 - 0.05x = 332  

< ==> -0.05x = -8 ==> x = 160, the first number ==> 400 - x = 240, the second number.