Inferences & Claims From Statistics - SAT Math

1. What is a skewed distribution?

   A distribution that is not symmetric. Data is pulled more toward one tail than the other.

2. What is a statistical inference?

   A conclusion about a population based on a sample. Uses sample data to make generalizations about the entire population.

3. What is the interquartile range (IQR)?

   IQR = Q3 - Q1. Measures the spread of the middle 50% of data values.

4. What is the range of the data set: 3, 6, 9, 12, 15?

   The range is 12. $15 - 3 = 12$; difference between largest and smallest values.

5. What does a histogram display?

   The frequency distribution of a data set. Shows how often different values or ranges occur in data.

6. Define 'population' in statistics.

   The entire set of individuals or items of interest. All possible subjects or objects being studied in research.

7. What does correlation measure?

   The strength and direction of a linear relationship. Values closer to -1 or 1 indicate stronger linear relationships.

8. Calculate the confidence interval: mean = 50, SD = 10, n = 25, 95% CI.

   The confidence interval is (46.08, 53.92). $50 \pm 1.96 \times \frac{10}{\sqrt{25}} = 50 \pm 3.92$.

9. Calculate the z-score for $x = 85$, $\text{mean} = 80$, $\text{SD} = 5$.

   The z-score is 1. $(85-80)/5 = 1$; standardizes the value relative to the distribution.

10. What is a box plot?

    A graphical summary of data distribution using quartiles. Shows median, quartiles, and potential outliers in data distribution.

11. Define 'sample' in statistics.

    A subset of the population used for analysis. Representative portion selected from the larger population for study.

12. Identify the median of the following numbers: 5, 2, 9, 4, 7.

    The median is 5. Sort values first: 2, 4, 5, 7, 9; middle value is 5.

13. What is the mode of the data set: 4, 4, 6, 8, 8, 8?

    The mode is 8. The value 8 appears most frequently (3 times).

14. What is the purpose of a confidence interval in statistics?

    Estimate range for population parameter. Provides likely bounds where true population value falls with given confidence.

15. Find and correct the claim: 'The sample result proves the hypothesis.'

    Correct: 'The sample result supports the hypothesis.'. 'Supports' is proper; samples provide evidence but never absolute proof.

16. What type of claim is made when stating a cause-and-effect relationship?

    Causal claim. Asserts one variable directly influences or causes changes in another.

17. What is the central limit theorem?

    The distribution of sample means approximates normality. For large samples, sample means follow a normal distribution pattern.

18. What does a p-value indicate in hypothesis testing?

    Probability of observed result under null hypothesis. Lower p-values suggest stronger evidence against the null hypothesis.

19. What is sampling bias?

    Bias due to non-random sample selection. Systematic error that makes samples unrepresentative of the population.

20. Identify the claim in the statement: 'The average height is over 6 feet.'

    The claim is that the mean height > 6 feet. States that the population mean exceeds 6 feet.

21. Which statistical measure is most affected by outliers?

    Mean. Extreme values pull the mean away from the center more than median or mode.

22. What is a parameter in statistics?

    A numerical characteristic of a population. Fixed value describing an entire population (usually unknown).

23. Define 'null hypothesis' in statistics.

    A statement that there is no effect or difference. The baseline assumption that researchers try to reject or fail to reject.

24. Identify the stronger correlation: $r = 0.7$ or $r = -0.7$.

    Both are equally strong. Both have the same absolute value, so equal strength.

25. Choose the correct alternative hypothesis: $H_0: \text{mean} = 50$

    $H_a: \text{mean} \neq 50$. Alternative hypothesis claims the parameter is not equal to the null value.

26. What is a Type II error in hypothesis testing?

    Failure to reject a false null hypothesis. Accepting a null hypothesis when the alternative is actually true.

27. What does 'margin of error' describe?

    The range of error in sample estimate of population. Quantifies the uncertainty in survey or poll results.

28. What is the critical value for a 95% confidence interval with a z-distribution?

    The critical value is 1.96. Marks the boundary for 95% of the standard normal distribution.

29. Which graph type is best for displaying frequency distributions?

    Histogram. Shows distribution shape and frequency of values across intervals or bins.

30. Find the range of the data set: 2, 8, 15, 22.

    Range = 20. Difference between maximum and minimum: $22 - 2 = 20$.

31. What does a 95% confidence interval mean?

    95% of samples contain the true population parameter. Indicates the reliability of the interval estimation method.

32. What is the definition of a statistical inference?

    Drawing conclusions about a population based on a sample. Uses sample data to make educated guesses about the entire population.

33. What is the correlation coefficient range?

    From -1 to 1. Perfect positive correlation is 1, perfect negative is -1.

34. Identify the formula for calculating the median of a data set.

    Arrange data, find middle value(s). For odd $n$, middle value; for even $n$, average of two middle values.

35. What is the mean of the data set: 3, 7, 8, 10, 15?

    The mean is 8.6. Sum all values and divide by count: $(3+7+8+10+15)/5 = 43/5 = 8.6$.

36. What is a normal distribution?

    A bell-shaped distribution symmetric about the mean. Most data falls within a predictable range around the center.

37. State the formula for calculating the range of a data set.

    Range = Maximum - Minimum. Range measures the spread by finding the difference between extremes.

38. State the formula for standard deviation in a sample data set.

    $s = \sqrt{\frac{1}{n-1} \sum (x_i - \bar{x})^2}$. Measures spread by calculating square root of variance with $n-1$ denominator.

39. Identify the null hypothesis in a statistical test.

    Null hypothesis: no effect or difference. Default assumption that treatment has no impact; what we test against.

40. What is the formula for variance?

    Variance = $\frac{\text{sum of squared deviations}}{n-1}$. Measures how spread out data points are from the mean.

41. What is the mode of the data set: 3, 3, 4, 5, 6?

    Mode = 3. The value that appears most frequently in the data set.

42. What is statistical significance?

    Results unlikely due to chance, given a threshold. Indicates results are unlikely to occur by random chance alone.

43. What is the mean of the data set: 4, 5, 7, 10?

    Mean = 6.5. Sum all values and divide by count: $(4+5+7+10) ÷ 4 = 6.5$.

44. What is a scatter plot?

    A graph showing the relationship between two variables. Each point represents one observation with two variable values.

45. What is a statistic in statistics?

    A numerical characteristic of a sample. Calculated value describing a sample (used to estimate parameters).

46. Identify the outlier in the set: 2, 4, 4, 5, 100.

    The outlier is 100. Value that falls far outside the typical range of the data.

47. Identify the term: The likelihood that a result is due to chance.

    P-value. Measures probability that observed results occurred by random chance alone.

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