The 116th House of Representatives of the United States of America has 435 members, of which 106 are women. An alien lands near the U.S. Capitol and treats members of the House as a random sample of the human race. He reports to his superiors that a 95% confidence interval for the proportion of the human race that is female has a lower bound of 0.203 and an upper bound of 0.284. What is wrong with the alien’s approach to estimating the proportion of the human race that is female? 

Verified step by step guidance

1. Step 1: Understand the context of the problem. The alien treats the 435 members of the House of Representatives as a random sample of the entire human race, which is a key assumption in constructing confidence intervals for population proportions.
2. Step 2: Recognize that the House of Representatives is not a random sample of the human race. It is a specific group selected based on political, demographic, and legal criteria, and it only represents a small subset of the U.S. population, not the global population.
3. Step 3: Recall the conditions for constructing a valid confidence interval for a population proportion. One important condition is that the sample must be randomly selected from the population of interest to ensure representativeness and unbiased estimation.
4. Step 4: Identify the error in the alien’s approach: using a non-random, non-representative sample (the House members) to estimate the proportion of females in the entire human race violates the random sampling assumption, making the confidence interval invalid for that purpose.
5. Step 5: Conclude that the confidence interval calculated from the House members' data can only be interpreted as an estimate of the proportion of females in that specific group, not the entire human race.

Key Concepts

Sampling Bias

Sampling bias occurs when the sample is not representative of the population being studied. In this case, the 116th House of Representatives is a specific group with selection criteria, not a random sample of the entire human race, so conclusions about the global female proportion based on this group are invalid.

Confidence Interval Interpretation

A confidence interval estimates a population parameter based on a random sample. It assumes the sample is representative and randomly selected. Using a confidence interval from a biased or non-random sample leads to misleading or incorrect inferences about the population.

Population vs. Sample

The population is the entire group of interest, while a sample is a subset used to make inferences about the population. The House of Representatives is a small, non-random subset of humans, so treating it as a sample of the entire human race ignores differences in demographics and selection processes.