Question

Answer each of the 4 part question using full sentences and good grammar. Use your own words.

A. List the conditions necessary for the CLT to hold. Make sure to list alternative conditions for when we know the population distribution is normal vs. when we don't know what the population distribution is, and the when the sample size is barely over 30 vs. when it's very large.

B. Explain, in plain English, the difference between standard error and margin of error.

C. Suppose a researcher is interested in evaluating the following claim ``The proportion of adults in the US who vote is 40%", and that she believes this is an underestimate.

c1. How should she set up her hypotheses?

c2. Explain to her, in plain language, how she should collect data and carry out a hypothesis test.

D. describe the differences and similarities between the hypothesis testing procedure using simulation and using theory. Especially discuss how the calculation of the p-value changes while the definition stays the same.

Answer 1

(a) Assumptions and Conditions

• Randomization Condition: The data must be sampled randomly.
• Independence Assumption: The sample values must be independent of each other.
• 10% Condition: When the sample is drawn without replacement (usually the case), the sample size, n, should be no more than 10% of the population.
• Sample Size Assumption: The sample size must be sufficiently large. In general, a sample size of 30 is considered sufficient if the sample is unimodal (and meets the 10% condition).

(b) Both characterize the quality of approximation of some unknown constant with the results of its statistical evaluation.
Knowing one of them, we can determine another.

(c) The hypothesis being tested is:

H₀: p = 0.40

H_a: p > 0.40

Collect a random sample of the adults in the US who vote.

(d) Hypothesis tests allow us to take a sample of data from a population and infer about the population.

Computer-based methods using randomization, simulation, and bootstrapping have much fewer restrictions.