Question 1: Calculate the probability of selecting a student with glasses from a class of 30 students, where 12 students wear glasses, using a random sample of 5 students.

Answer: 0.4

Question 2: In a bag with 50 marbles (20 red, 30 blue), what is the expected number of red marbles in a random sample of 10 marbles?

Answer: 4

Question 3: A surveyor randomly selects 15 houses from a street of 150 houses. Explain why this is an example of random sampling.

Answer: Because each house has an equal chance of being selected, ensuring unbiased representation.

Question 4: Plot the graph of y = 2x for x values from 0 to 10 on the provided grid.

Answer: See full answer key

Question 1: A school has 200 students, of whom 80 are in the choir. If 25 students are randomly selected for a survey, what is the expected number of choir members in the sample?

Answer: 10

Question 2: Identify and correct the mistake in this statement: 'Random sampling always produces a sample that perfectly represents the population.'

Answer: Random sampling reduces bias but may not always produce a perfect representation.

Question 3: Construct a simple random sample of size 4 from the numbers 1 to 20 using the grid.

Answer: See full answer key

Question 4: In a study, a researcher randomly selects 30 people out of 300. The researcher claims this sample is representative. What factors should be considered to verify this claim?

Answer: Sample size, sampling method, population variability, and whether all groups are proportionally represented.

Question 1: A factory produces 500 items daily, with a defect rate of 2%. If a random sample of 20 items is tested daily, what is the probability that exactly 1 defective item is found? Use the Poisson approximation if appropriate.

Answer: 0.2707

Question 2: Explain why increasing the sample size in a random sampling process can lead to more accurate representation of the population. Support your reasoning with statistical concepts.

Answer: Larger samples tend to better reflect the population, reducing sampling error and increasing reliability.

Question 3: A researcher conducts two independent simple random samples of sizes 50 and 100 from the same population. Discuss how the variance of the sample means compares and why.

Answer: The variance of the mean from the larger sample (n=100) is smaller, because variance decreases with larger sample size.