Question

Let p^1p^1 be the proportion of the 30 Americans sampled that agree climate change is an immediate threat to humanity.

Suppose that the environmental group will repeat this study again, but with 100 American citizens sampled. Let p^2p^2 be the proportion of these 100 Americans sampled that agree climate change is an immediate threat to humanity.

Which of the following statements is true? (Again, assume the proportion of all Americans that agree climate change is an immediate threat to humanity is 0.35.)

Select one:

a. p^1p^1 is more likely to be greater than 0.3 than p^2p^2 is.

b. p^1p^1 is more likely to be greater than 0.35 than p^2p^2 is.

c. p^1>p^2p^1>p^2

d. p^2p^2 is more likely to be greater than 0.3 than p^1p^1 is.

Answer 1

d. p^2p^2 is more likely to be greater than 0.3 than p^1p^1 is.

Explanation:

We have two samples: one with n1=30 and a proportion p1, and the other with n2=100 and proportion p2.

We also know that the population is π=0.35.

As the sample size increase, the spread in the estimation of the true proportion is reduced. This is because the standard deviation of the sampling distribution decrease when the sample size increase.

Then, we would expect p2 to be closer to the true proportion than p1.

This is the only thing we can claim: p2 is more likely to be closer to the true population than p1.

Then, from the options available, we have:

a. p^1p^1 is more likely to be greater than 0.3 than p^2p^2 is.

FALSE. p2 has an expected spread over the true population that is less than p1, so it is expected to be more close to 0.35, therefore bigger than 0.3, than p1.

b. p^1p^1 is more likely to be greater than 0.35 than p^2p^2 is.

FALSE. We can not claim that, because 0.35 is the population proportion and p1 and p2 can fall at either side of this value with equal probability.

c. p^1>p^2p^1>p^2

FALSE. p1 can be higher or lower than p2.

d. p^2p^2 is more likely to be greater than 0.3 than p^1p^1 is.

TRUE. As p2 is expected to be around the true proportion with less spread, is more likely to be greater than 0.3 than p1. It also will be more likely to be under 0.4 than p1.