Introducing Statistics: Learning from Data - AP Statistics

1. What is the empirical rule?

   The empirical rule states that 68%, 95%, and 99.7% of data fall within 1, 2, and 3 standard deviations from the mean, respectively. Applies specifically to normal distributions with predictable percentages.

2. How is variance calculated in a sample?

   Variance = $\frac{\text{sum of squared deviations from mean}}{n-1}$. Uses $n-1$ for sample variance to correct for bias.

3. What is a residual in regression analysis?

   A residual is the difference between an observed value and the value predicted by a model. Shows how far off the model's prediction was from reality.

4. Define standard deviation.

   Standard deviation is the square root of the variance. Provides a measure of spread in the same units as the data.

5. What is a z-score?

   A z-score measures how many standard deviations an element is from the mean. Standardizes values for comparison across different datasets.

6. Define inferential statistics.

   Inferential statistics make predictions or inferences about a population based on sample data. Uses sample data to draw conclusions about the entire population.

7. Define independent events in probability.

   Two events are independent if the occurrence of one does not affect the probability of the other. Knowledge of one event provides no information about the other.

8. Identify the significance level in hypothesis testing.

   The significance level is the probability of a Type I error, typically denoted by $\text{alpha}$. Sets the threshold for rejecting the null hypothesis.

9. What does 'mutually exclusive' mean in probability?

   Two events are mutually exclusive if they cannot occur at the same time. No overlap possible; one event prevents the other.

10. What is a confidence interval?

    A confidence interval is a range of values that is likely to contain a population parameter. Provides a range estimate with a specified level of confidence.

11. Identify the purpose of a box plot.

    A box plot visually displays the distribution and identifies outliers of a data set. Shows five-number summary and highlights unusual values.

12. Define mode in terms of a data set.

    The mode is the value that appears most frequently in a data set. Identifies the most common or repeated value in the dataset.

13. How is relative frequency calculated?

    Relative frequency is calculated as the frequency of an event divided by the total number of observations. Converts counts to proportions for easier comparison.

14. What is a sample in the context of statistics?

    A sample is a subset of the population from which data is actually collected. A smaller group selected from the population for practical data collection.

15. What is meant by extrapolation?

    Extrapolation involves predicting beyond the range of observed data. Making predictions outside the observed data range.

16. What is the main goal of descriptive statistics?

    To summarize or describe the characteristics of a data set. Focus is on organizing and presenting data, not making predictions.

17. How do you calculate the probability of two independent events occurring together?

    Probability = P(A) × P(B) for independent events A and B. Multiply individual probabilities when events don't influence each other.

18. What does the central limit theorem state?

    The central limit theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases. Sample means become normally distributed regardless of population shape.

19. What is a Type I error?

    A Type I error occurs when the null hypothesis is incorrectly rejected. A false positive; rejecting a true null hypothesis.

20. What is the median of a data set?

    The median is the middle value when data points are arranged in order. Half the values are above and half are below this central value.

21. Identify the formula for the line of best fit.

    Line of best fit: $y = mx + b$, where $m$ is the slope, $b$ is the y-intercept. Standard linear equation form for the regression line.

22. What is a histogram?

    A histogram is a graphical representation of the distribution of numerical data. Shows frequency of data values using bars or bins.

23. What is the purpose of a residual plot?

    A residual plot is used to assess the fit of a regression model. Helps identify patterns in errors and model appropriateness.

24. What is meant by the term 'skewness' in a data set?

    Skewness describes asymmetry in the distribution of values in a data set. Indicates whether data leans left or right from center.

25. What is a null hypothesis in statistics?

    A null hypothesis is a statement that there is no effect or no difference, used as a starting point. Assumes no relationship exists; the default position to test against.

26. What is the purpose of a contingency table?

    A contingency table displays the frequency distribution of variables. Cross-tabulates two categorical variables to show relationships.

27. What is a sampling distribution?

    A sampling distribution is the probability distribution of a statistic obtained from a large number of samples. Shows how sample statistics vary across repeated sampling.

28. Define correlation coefficient.

    The correlation coefficient measures the strength and direction of a linear relationship. Ranges from -1 to +1, indicating weak to strong relationships.

29. What does range measure in a data set?

    Range measures the difference between the maximum and minimum values. Simple measure of spread calculated as max - min.

30. What is a normal distribution?

    A normal distribution is a bell-shaped distribution that is symmetric about the mean. The classic bell curve with equal tails on both sides.

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

    A p-value indicates the probability of observing the test results under the null hypothesis. Measures how likely our results are if the null hypothesis is true.

32. State the formula for calculating the mean of a data set.

    Mean = $\frac{\text{sum of all data points}}{\text{number of data points}}$. Add all values and divide by the count to find the average.

33. Define categorical variable.

    A categorical variable is a variable that represents categories or groups. Takes on distinct labels or names rather than numerical values.

34. What does a correlation coefficient of 0 indicate?

    A correlation coefficient of 0 indicates no linear relationship between variables. The variables move independently with no linear pattern.

35. What is an outlier in statistics?

    An outlier is a data point that differs significantly from other observations. An unusual value that stands apart from the typical pattern.

36. What is the definition of a population in statistics?

    A population is the entire group of individuals or instances about whom we hope to learn. The complete group we want to study and make conclusions about.

37. What is a Type II error?

    A Type II error occurs when the null hypothesis is not rejected when it is false. A false negative; failing to reject a false null hypothesis.

38. What is a continuous variable?

    A continuous variable can take an infinite number of values within a given range. Can be measured to any desired precision within its range.

39. What is a scatter plot used for?

    A scatter plot is used to show the relationship between two quantitative variables. Each point represents one observation with two measurements.

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