Dear everyone

With all due respect, may I ask a help for some O/X quiz?

1)Suppose two random variables X and Y are negatively correlated. If X is above its mean, then Y is always below its mean.
(I know that when two are negatively correlated, if X gose up, Y goes down. However, I am not sure about the relation with mean)

2)Suppose X is a binomial random variable. Then 1000X is approximately a normal random variable.
(I know when n is bigger, it would be a normal distribution. However, I am not sure about 1000X..)

3)The polling agency should take a larger sample if the population is larger.
(I think that sample would be not affect whether the population is larger.. but not sure)

4)Linear regression framework can be applied to capture the decreasing marginal returns between two variables.
(I think it is O)

1) If X is at its mean then Y is at its mean. Like you say when X goes up from its mean then Y will go down from its mean.

2) 1000X is not normally distributed. If X's population was {1,0,0,1,0,1} then 1000X's population would be {1000,0,0,1000,0,1000} it still only has two values.

3) We correct for things such as standard error for a finite population with a "finite correction factor" $\sqrt{\frac{N-n}{N-1}}$ (Where N is population size and n is sample size) if N increases then you do need to increase n to maintain the same accuracy.

4) Decreasing marginal returns tends asymptotically to zero. You cannot use a linear relationship to model that.

As for the question 1, the questions says 'always', so can that be false, because a value of Y can sometimes be above its mean even if a value of X is also above its mean?

Have a great day.

y will always be below its mean when increasing x. If the slope of the regression line has a slope m which is negative then when x increases by an amount d, y will change by an amount md. Since m is negative md is negative and y decreases.