Review Unit 2

Main ideas and concepts covered

1) Quantitative vs. categorical (qualitative) variables

• Quantitative variables

  • Are numbers where arithmetic operations make sense.
  • If it makes sense to compute an average/mean, the variable is quantitative.
  • Examples:
    • Heights (average height)
    • Time (average time)
    • Ages (average age)
    • Recent test scores (average score)

• Categorical variables

  • Put observations into categories; typically represented as counts or proportions.
  • Examples:
    • ZIP codes (not meaningful to average)
    • Gender (categories: e.g., male/female)
    • Social Security number groups (e.g., based on ranges of the last digits)

• Terminology note

  • College Board uses “categorical” (some books say “qualitative”).

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2) Using the right type of statistical test for the data

• Categorical data → work with proportions/percentages

  • Tests: z-tests and chi-square tests
  • Examples:
    • One-proportion z test (“one prop z”)
    • Two-proportion z test (“two prop z”)
    • Chi-squared tests:
      • Goodness of fit
      • Homogeneity
      • Independence

• Quantitative data → work with means/averages

  • Use t/z/linear regression procedures depending on the situation
  • Examples:
    • One-sample t
    • Two-sample t
    • Paired t
    • Linear regression (mentioned as part of the “t family” context)
  • Note:
    • Z procedures listed were proportion-related (one prop z, two prop z)
    • Chi-squared remains categorical

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3) One-variable distributions: center/spread and checking shape

When analyzing a distribution, check:

• B: shape
• S: spread
• Center

Key emphasis:

• Use context for center/spread.
• The mean is not resistant to outliers (it can be pulled toward extremes).
• The median is more resistant.

Effect of outliers (illustrated):

• With a large high outlier: mean > median (mean pulled right)
• With a large low outlier: mean < median (mean pulled left)
• Without extreme impact: mean and median can be the same or close

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4) Graph types for one-variable data

• Frequency graph

  • Like a histogram, but framed as frequencies

• Relative frequency graph

  • Bar heights are percentages
  • Percentages must sum to 100%

• Cumulative frequency graph

  • Also called an ogive
  • Can be cumulative relative frequency
  • Starts at 0 and increases to 100%

Reading quantiles from an ogive:

• Median: at 50%
• Q1 (25th percentile): at 25%
• Q3 (75th percentile): at 75%
• IQR = Q3 − Q1

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5) Standard deviation: calculator method and interpretation

Calculator method (example workflow):

• Input data (example values): 23, 36, 21, 40, 27
• Use a path like:
  • Menu → Statistics → Calculations → 1-Var Stats
• Put the list in X list
• Read the standard deviation output
  • (Rounding may differ from hand calculations.)

Transformation facts (linear transformation of data):

If you transform data via y = a + b(x):

• Add a constant (e.g., +8):
  • Shifts the mean but does not change standard deviation
• Multiply by a constant (e.g., ×2):
  • Rescales the spread
  • Standard deviation multiplies by |b|

Combining independent variables (mean and SD/variance relationship):

• Mean of sum:

  • [ \text{mean}(a+b)=\text{mean}(a)+\text{mean}(b) ]

• Standard deviation of sum:

  • handled through variances (variance addition)
  • speaker states you add variances, not SDs (variance addition rule implied)

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6) Normal distribution: empirical rule and Normal CDF / inverse Normal

Empirical rule (68–95–99.7)

• 68% within 1 standard deviation of the mean
• 95% within 2 standard deviations
• 99.7% within 3 standard deviations

Normal “four types” of problem setup

1. Probability of form P(X > k) / P(X < k) using normalcdf
2. Probability between two values P(a < X < b) using normalcdf
3. Find a cutoff for a given left-tail probability using inverse normal
4. Find an x-value for a given percentile using inverse normal

Workflow (z-score concept + calculator functions)

• Convert conceptually to z-scores: [ \frac{x-\mu}{\sigma} ]

• Use calculator functions:

  • normalcdf(lower, upper, mean, sd) for probabilities
  • invNorm(areaLeft, mean, sd) for quantiles

Example logic:

• Movies per semester modeled as normal with:
  • mean = 5
  • sd = 2
• Use:
  • P(X > 6): normalcdf(6, ∞, 5, 2)
  • P(3 < X < 8): normalcdf(3, 8, 5, 2)
  • P(X < 2): normalcdf(-∞, 2, 5, 2)
• Bottom 15% cutoff:
  • inverse normal with left-tail area = 0.15

Z-score comparison example:

• Test A: mean 83, sd 3
• Test B: mean 84, sd 5
• Student scores:
  • Test A: 79
  • Test B: 90

• Compute z-scores:

  • [ z_A=\frac{79-83}{3} ]

  • [ z_B=\frac{90-84}{5} ]

• Interpretation:

  • The larger absolute z-score is more unusual / farther from the mean
  • Conclude the score with the larger distance from the mean is better relative to its test distribution.

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7) Two-variable data (bivariate): regression and correlation

Definitions:

• Explanatory variable: X
• Response variable: Y

• Regression line:

  • [ \hat{y}=b_0+b_1x ]

  • (\hat{y}) = predicted response

  • (b_1) = slope (change in predicted (y) per unit (x))
  • (b_0) = y-intercept (predicted (y) when (x=0))

Scatterplot interpretation:

• Outliers
  • Case A: outlier not following the pattern
  • Case B: influential point
    • can pull the regression line
    • can strengthen/alter the slope

Least squares regression idea:

• Residuals = observed − predicted
• Least squares line minimizes the sum of squared residuals

Residual sign interpretation (multiple choice):

• Residual > 0 (observed > predicted): prediction was too low
• Residual < 0 (observed < predicted): prediction was too high

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8) Checking regression assumptions: residual plots

Recommendations:

• Make a residual plot
• Look for random scatter around 0
• If a pattern appears, the model may not be appropriate

AP note:

The AP test may ask you to evaluate patterns; “AP test does that for you,” but you still must interpret the resulting straightness/pattern.

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9) Interpreting regression output: slope, intercept, r, and r²

Example regression output:

• [ \hat{y}=1.2x+31 ]

• Slope (1.2)

  • For each 1-degree increase in X (Denver temperature), predicted Y (Flat Top Mountain temperature) increases by 1.2 degrees
• Intercept (31)
  • When (X=0), predicted (Y) is 31 (as a prediction/average statement)

Correlation coefficient (r)

• Described as indicating positive linear correlation
• Direction and strength:
  • Direction: positive vs negative
  • Strength: strong/moderate/weak (depending on whether the prompt uses (r) or (r^2))

Coefficient of determination (r²)

• Interpreted as:
  • the percent of variation in Y explained by the regression model (least-squares line)

Extrapolation vs. interpolation:

• Using regression outside the data domain = extrapolation (warned against)

Correlation vs causation:

• Correlation doesn’t imply causation:
  • possible lurking variables / confounding
• To claim causation:
  • need a well-designed controlled experiment with randomization

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Speakers / sources featured

• No named speakers or external sources are explicitly identified in the subtitles.
• The content is presented by an instructor/teacher (unnamed).