Sampling Basics
Define a random sample and explain why it's important when gathering data.
Answer: A random sample is when every member of the population has an equal chance of being selected. It's important because it helps make sure the sample is representative and gives you reliable information about the population.
What does it mean for a sample to be representative?
Answer: A representative sample has the same characteristics and distribution as the population, so it gives an accurate estimate of the population.
What does the Mean Absolute Deviation (MAD) tell you about a data set?
Answer: The MAD tells you how spread out the data is from the mean. A larger MAD means the data is more spread out.
What is the Interquartile Range (IQR) and what does it measure?
Answer: The IQR is the difference between the third quartile and first quartile. It measures the spread of the middle 50% of the data.
If 8 out of 12 customers surveyed said they liked the new menu item, what proportion liked it?
Answer: 8/12 = 2/3, or about 67%
Marcus wants to know the favorite lunch in his school. He surveys every 5th student as they leave the cafeteria. Is this a random sample? Why or why not?
Answer: No, it's not a random sample because not every student has an equal chance of being picked (only students leaving at the right time would be selected).
A store manager wants to know how long customers spend shopping. The population shows most customers spend 15 minutes at the store. Which sample would NOT be representative: (A) 5 customers with times of 12, 15, 18, 14, 16 minutes or (B) 5 customers with times of 45, 50, 42, 48, 46 minutes?
Answer: (B) would NOT be representative because the times are much higher than the population average of 15 minutes.
Two classes took the same test. Class A has a mean of 75 and MAD of 3. Class B has a mean of 75 and MAD of 8. Which class had more consistent test scores?
Answer: Class A has more consistent scores because it has a smaller MAD (3 vs 8), meaning the scores are closer to the mean.
Comparing two data sets: Dataset 1 has IQR = 10, and Dataset 2 has IQR = 25. Which has greater variability in its middle 50%?
Answer: Dataset 2 has greater variability because its IQR is larger (25 vs 10).
A survey shows 30% of shoppers prefer Brand A. If there are 500 total shoppers, how many would you estimate prefer Brand A?
Answer: 0.30 × 500 = 150 shoppers
What is the population and what is a sample in this situation: A teacher measures the height of 15 students in a class to estimate the average height of all seventh graders in the school?
Answer: The population is all seventh graders in the school. The sample is the 15 students in the class.
If the population of students at a school is 60% girls and 40% boys, approximately what should a representative sample of 100 students look like?
Answer: The sample should have about 60 girls and 40 boys to match the population proportions.
If a sample has mean height of 60 inches and MAD of 2 inches, what does this tell you about how the heights are distributed?
Answer: Most heights are within 2 inches of 60 inches, meaning heights are clustered close to the mean (generally between 58 and 62 inches).
A school compares test scores from two different classes using: Class X: Mean = 80, MAD = 5 and Class Y: Mean = 80, IQR = 15. Based on MAD and IQR, which class has more variability?
Answer: Class Y likely has more variability because the IQR of 15 suggests a wider spread than the MAD of 5.
Larger samples are generally better than smaller samples. How does increasing sample size help reduce sampling error?
Answer: A larger sample gives you more data points, which makes it more likely that your sample mean will be close to the true population mean.
True or False: A random sample will always have the exact same mean as the population mean?
Answer: False. A random sample's mean is likely to be close to the population mean, but it probably won't be exactly the same.
A researcher surveys 20 people at a gym about their exercise habits. Would this sample be representative of exercise habits for all people in the city? Explain why or why not.
Answer: No, because the sample only includes people who go to the gym, which doesn't represent people who don't exercise or exercise differently.
A data set has a mean of 100 and a MAD of 0. What can you say about all the values in the data set?
Answer: All values must be exactly 100 (no variation at all). When MAD is 0, all data points are identical.
Two datasets have the same median of 50. Does this mean they have the same variability? Explain.
Answer: No, median alone doesn't tell you about variability. You need to look at MAD, IQR, or range to compare spread.
A pet store surveys customers about pet ownership by asking visitors at the store. Explain why this sample might be biased and how to fix it.
Answer: The sample is biased because people at a pet store already own or are interested in pets. To fix it, randomly survey people from the general population (like shopping malls, neighborhoods, etc.).
When comparing two groups of data, what should you consider: the medians, the variability, both, or something else?
Answer: You should consider both the center (mean or median) and the variability (MAD, IQR, range) to get a complete picture of how the groups compare.
If Group A has a mean of 50 and Group B has a mean of 48, is there definitely a meaningful difference between the groups? Why or why not?
Answer: Not necessarily. You need to look at the variability (MAD or IQR) to see if the 2-point difference is significant compared to the spread of each group.
Group A data: 10, 12, 11, 13, 12. Group B data: 5, 15, 10, 20, 10. Both have the same mean of about 12. Why is Group A's data more reliable for prediction?
Answer: Group A is more reliable because it has less variability (all values close to 12), so predictions based on the mean are more likely to be accurate. Group B has much more spread.
A sample shows 40% of people support a new policy. Would you expect exactly 40% of the entire population to support it? Explain.
Answer: No, because sampling variation means the true population proportion is probably close to 40%, but unlikely to be exactly 40%.
Two samples: Sample 1 has mean 60 and MAD 15. Sample 2 has mean 65 and MAD 2. The samples are different sizes. What does the smaller MAD in Sample 2 tell you?
Answer: The values in Sample 2 are much closer together and more consistent around the mean than in Sample 1.