Topic Review — Sampling and Proportions
This review covers all lessons in this topic: random sampling and bias, sample proportions as random variables, and the distribution of sample proportions.
Review Questions
- Explain the difference between a population parameter and a sample statistic. Give one example of each.
- A survey of 400 voters finds 220 support a proposed law. Calculate the sample proportion p̂ and explain what it estimates.
- In a random sample of size n from a population with true proportion p, state the mean and standard deviation of the sample proportion p̂.
- For p = 0.4 and n = 100, find: (a) E(p̂) (b) Var(p̂) (c) SD(p̂).
- State the two conditions that must be met for the distribution of p̂ to be approximately normal. Why are these conditions needed?
- A coin is suspected of being biased. In 200 flips, 112 heads appear. Check the normality conditions (using p = 0.5) and state the approximate distribution of p̂.
- Using the distribution from Q6, find P(p̂ > 0.56).
- Explain what “sampling variability” means. Why does increasing n reduce sampling variability?
- A large school has 35% of students who walk to school. Random samples of size n = 200 are repeatedly taken.
- (a) Find the mean and standard deviation of the distribution of p̂.
- (b) Find the probability that a sample proportion falls between 0.30 and 0.40.
- (c) Find the value c such that P(p̂ > c) = 0.025.
- A national survey finds that 68% of adults own a smartphone. A journalist takes a random sample of 150 adults and finds 92 own smartphones.
- (a) Find the observed sample proportion p̂.
- (b) Find P(p̂ ≤ 92/150) given p = 0.68.
- (c) Is the journalist’s result unusual? Justify using probability.
- Without sampling: explain in one sentence why p̂ is called an unbiased estimator of p.
- A factory produces items, 8% of which are defective. Quality control takes samples of n = 400.
- (a) Verify normality conditions for p̂.
- (b) Find P(p̂ > 0.10).
- Describe a potential source of sampling bias for each of these scenarios:
- (a) Estimating average weekly screen time by asking students in the school library.
- (b) Estimating the proportion of adults who exercise daily using an online health forum.
- A polling organisation finds that in a sample of n = 600 voters, p̂ = 0.52 support Candidate A.
- (a) Find P(p̂ ≥ 0.52 | p = 0.50) — i.e. the probability of this result or more extreme if the race is tied.
- (b) Does this provide strong evidence that Candidate A is ahead? Justify.
- Suppose n = 150 and p̂ = 0.48. Explain what would happen to the distribution of p̂ if (a) n was increased to 1500 while p remained 0.48, and (b) p changed to 0.20 while n stayed at 150.