Draw Inferences From Random Samples - 7th Grade Math

1. Which sample is more likely to give a stable estimate: $n=20$ or $n=200$?

   $n=200$. Larger samples reduce sampling variability and give more stable estimates.

2. If many simulated samples give results between $0.47$ and $0.53$, what is a reasonable prediction interval?

   About $0.47$ to $0.53$. The interval shows expected sampling variability.

3. What is the formula for the mean of a sample with values $x_1, x_2, \dots, x_n$?

   $\bar{x}=\frac{x_1+x_2+\cdots+x_n}{n}$. Add all values and divide by count $n$.

4. Four simulated sample proportions are $0.52, 0.49, 0.55, 0.44$. What is the range?

   $0.55-0.44=0.11$. Range = highest - lowest value.

5. What is the sample proportion used for when estimating a population proportion?

   It is the estimate of the population proportion. Sample proportions estimate population proportions.

6. Four same-size samples give means $8, 9, 7, 10$. What is the range of these estimates?

   $3$. Range = maximum - minimum = $10 - 7 = 3$.

7. What is a simulation in the context of sampling?

   A repeated random process used to imitate taking many samples. Simulations model real sampling without actual data collection.

8. What is a statistic in this standard: population value or sample value?

   Sample value (computed from the sample). Statistics are calculated from samples, parameters describe populations.

9. Which estimate is more precise: a random sample of size $20$ or size $200$?

   Size $200$. Larger samples have less variability and more precision.

10. What is the main effect of increasing $n$ on variability of sample means?

    Variability decreases (estimates tend to be closer together). Larger samples produce more consistent estimates across trials.

11. A random sample predicts $52\%$ will vote for Candidate A. What is the predicted winner?

    Candidate A. With $52%$ support, Candidate A has the majority.

12. What is sampling variability?

    Natural differences in results from sample to sample. Different samples naturally give different results.

13. What is the population in a study that uses a sample to make an inference?

    The entire group of interest you want information about. The population includes everyone you want to study, not just the sample.

14. What is sampling bias?

    A method that systematically favors certain outcomes. Bias leads to estimates that consistently miss the true value.

15. A sample mean word length is $4.8$ letters. What is the best estimate of the population mean word length?

    $4.8$ letters. Sample mean is the best estimate of population mean.

16. A survey uses the first $30$ students entering school. Identify the likely issue: random sample or biased sample?

    Biased sample (not random). First arrivals may not represent all students.

17. What is a simulation in the context of sampling and inference?

    Using random processes to imitate repeated sampling. Simulations model real sampling to study variability.

18. Identify the population: 50 students are randomly surveyed about school lunch.

    All students at the school. The population is who we want to learn about, not just who we surveyed.

19. What is the population characteristic you are trying to estimate called?

    A parameter (an unknown characteristic of the population). Parameters describe the entire population.

20. Choose the best inference: sample mean word length is $4.8$, so the book's mean is about what value?

    About $4.8$ letters per word. Sample mean is the best estimate for population mean.

21. A random sample of $50$ students has $28$ who prefer pizza. What is the sample proportion $\hat{p}$?

    $\hat{p}=\frac{28}{50}=0.56$. Divide favorable outcomes by total sample size.

22. Two random samples of the same size give means $12$ and $18$. What is the absolute difference?

    $|18-12|=6$. Shows variability between two sample estimates.

23. What is the unknown characteristic of interest in sampling called?

    A population parameter. Parameters are population values we estimate from samples.

24. Which option describes an unbiased method: random sample or convenience sample?

    Random sample. Random sampling gives each member equal selection probability.

25. What is the sample mean used for when estimating a population mean?

    It is the estimate of the population mean. Sample statistics estimate corresponding population parameters.

26. Multiple same-size samples give proportions $0.48, 0.51, 0.49, 0.52$. About how far off might $0.50$ be?

    About $\pm 0.02$. Values vary from $0.48$ to $0.52$, about $0.02$ from center.

27. What is the sample mean word length if lengths are $4,5,3,8$?

    $5$. Mean = $(4+5+3+8) div 4 = 20 div 4 = 5$.

28. What is a sample in a sampling study?

    A subset of the population that is actually measured. Samples are smaller groups used to make inferences about the whole population.

29. Identify the sample: 50 students are randomly surveyed about school lunch.

    The 50 surveyed students. The sample consists only of those actually surveyed.

30. If repeated random samples of size $n$ give proportions from $0.46$ to $0.54$, what is a reasonable margin of error?

    About $0.04$. Margin of error ≈ half the range = $(0.54-0.46) div 2 = 0.04$.

31. What usually happens to sampling variability when sample size increases?

    It decreases. Larger samples better represent populations, reducing variation.

32. What is sampling variability?

    Natural differences in estimates from different random samples. Different samples give different results due to chance alone.

33. Find the sample mean of $4, 6, 10, 10$.

    $7.5$. Mean = $(4+6+10+10) div 4 = 30 div 4 = 7.5$.

34. Identify the more likely biased method: survey first $30$ students entering school or random list?

    Survey the first $30$ students entering school. Early arrivals may differ systematically from the whole population.

35. Find the range of sample means for $4.8,5.1,5.0,4.9$.

    $0.3$. Range = max - min = $5.1 - 4.8 = 0.3$.

36. What is a sample in a statistical study?

    A subset of the population used to collect data. Samples represent the population but are smaller and manageable.

37. What does it mean for a sample to be random?

    Every member has an equal chance to be selected. Random selection ensures no systematic bias in choosing members.

38. What is the sample proportion if $18$ of $30$ students choose Candidate A?

    $0.6$. Proportion = $\frac{18}{30} = 0.6$ chose Candidate A.

39. What is a population in a statistical study?

    The entire group being described or studied. The population includes everyone/everything you want to study.

40. What happens to sampling variability when the sample size increases?

    It usually decreases. Larger samples give more stable estimates.

41. What is the predicted number in a population of $800$ if $\frac{1}{4}$ of a sample has a trait?

    $200$. $\frac{1}{4} \times 800 = 200$ expected to have the trait.

42. Which option is the best description of sampling variability: changes in results across random samples or mistakes in math?

    Changes in results across random samples. Variability is natural variation, not errors.

43. What is a parameter in this standard: population value or sample value?

    Population value (the true but usually unknown value). Parameters are fixed but unknown; we estimate them using statistics.

44. In a school election sample, $60\%$ choose Candidate A. What is the best prediction for the winner?

    Candidate A. Majority in sample predicts population winner.

45. What is the main purpose of taking multiple random samples of the same size?

    To measure variability in the estimate from sample to sample. Multiple samples show how much estimates can vary.

46. Find the sample proportion if $18$ out of $30$ surveyed students say "yes."

    $0.6$. Proportion = favorable outcomes $div$ total = $18 div 30$.

47. What is the sample size, written as $n$?

    The number of individuals or items in the sample. Sample size $n$ determines how many observations we collect.

48. What does it mean if sample results vary a lot across repeated samples?

    The estimate is less reliable (high sampling variability). High variability means less confidence in the estimate.

49. What is the sample statistic used to estimate a population characteristic called?

    An estimate (a sample statistic used to infer a population parameter). Sample statistics estimate population parameters.

50. Which method best reduces bias: random sampling or choosing only volunteers?

    Random sampling. Random selection gives everyone equal chance, reducing bias.

51. What is the population in a sampling study?

    The entire group you want information about. The population includes everyone you're studying, not just those measured.

Study these flashcards