Exam Review Notes

Contingency Tables

• When given a contingency table, it's essential to find the totals for rows and columns.

  • Example:
    • Rows: 30 + 27 = 57, 41, 35
    • Columns: 67, 66
    • Total Total: 133

• Calculating probability of male or no:

  • P(Male \cup No) = \frac{Total Male + Total No - (Male \cap No)}{Total Total}
  • P(Male \cup No) = \frac{67 + 41 - 18}{133} = 0.6767

Conditional Probability

• The slash notation (/) often indicates conditional probability.
• P(Yes | Female): Probability of 'Yes' given 'Female.'
  • Focus only on the 'Female' column.
  • P(Yes | Female) = \frac{27}{66}

Confidence Intervals

• Steps for confidence interval calculations:
  • Use calculator functions.
  • Example result (placeholders): CLower = 0.037, CUpper = 0.17

Regression Analysis

• Input data into calculator using 'stat' then 'regression'.

  • Two lists: one for x, one for y.

• Obtain regression equation:

  • y = mx + b
    • m: Slope
    • b: Y-intercept
  • Example: m = 0.6374, b = 4.1099
    • y = 0.6374x + 4.1099

• Residual Calculation:

  • Find predicted y (y-hat) by plugging x into the regression equation.
    • Example: x = 5, y = 0.6374 * 5 + 4.1099 = 7.2969 \approx 7.3
  • Residual = Actual y - Predicted y.
    • Example: y = 7, Residual = 7 - 7.3 = -0.3

• Coefficient of Determination:

  • R-squared value from calculator.
  • Example: R^2 = 0.711

Combinations and Permutations

• Permutation: Order matters (e.g., first, second, third place).

  • Example: Selecting first, second, and third place from 12 people.
    • P(n, k) = \frac{n!}{(n-k)!}, where n = 12 and k = 3
    • P(12, 3) = 1320

• Combination: Order does not matter (e.g., picking chocolates).

  • Example: Selecting 5 men out of 10 and 5 women out of 12.
    • C(n, k) = \frac{n!}{k!(n-k)!}, where n = 10, k = 5 for men, and n = 12, k = 5 for women.
    • C(10, 5) * C(12, 5) = 1,585,584

Binomial Distributions

• Using calculator for binomial problems.

• BinomPDF: Probability of exactly x successes.

  • Used when a question includes word "exactly".
  • n = number of trials
  • x = number of sucesses
  • p = probability of sucess
  • Example: 12 trials, exactly 3 successes, p = 0.66.
    • BinomPDF(12, 3, 0.66) = 0.00384

• BinomCDF: Cumulative probability (less than, greater than, or between).

  • Used when a question includes words like "less than", "greater than", or "between".
  • Requires lower and upper bounds.
    • Lower bound minimum: 0
    • Upper bound maximum: n
  • Example: 12 children, probability = 0.64, less than 3.
    • BinomCDF(12, 0.64, 0, 2) = 0.001094

Hypothesis Testing

• T-Test:

  • Used when the population standard deviation is unknown.

  • Tests a claim about a population mean.

  • Inputs:

    • \mu_0: Claimed mean.
    • \bar{x}: Sample mean.
    • s: Sample standard deviation.
    • n: Sample size.

  • Alternative Hypothesis: Specifies the direction of the test (less than, greater than, not equal to).

  • P-value: Probability of obtaining a test statistic as extreme as, or more extreme than, the one actually observed, assuming the null hypothesis is true.

    • If P-value < α (significance level), reject the null hypothesis. There is enough evidence.
    • If P-value > α, fail to reject the null hypothesis. There is not enough evidence.

Expected Value

• Expected value calculation:

  • (Amount gained * Probability of gain) - (Amount lost * Probability of loss).
  • Example: Life insurance cost is $700, $70,000 coverage, probability of staying alive is 0.9986.
    • Expected Value = (700 * 0.9986) - (70000 - 700) * (1 - 0.9986) ≈ -690.2

Normal Distribution

• NormalCDF: Used for finding probabilities in a normal distribution.

  • Inputs:

    • Mean (\mu).
    • Standard deviation (\sigma).
    • Lower bound.
    • Upper bound.

  • Example: Mean = 93.7, Standard deviation = 4.2, less than 40.

    • NormalCDF(\mu=93.7, \sigma=4.2, Lower Bound=-100000, Upper Bound=40) = 0

Confidence Intervals (Proportions)

• One-Prop Z-Interval: Used for estimating a population proportion.

  • Inputs:

    • x: Number of successes.
    • n: Sample size.
    • Confidence level.

  • Example: x = 67, n = 48, Confidence = 0.95

Confidence Intervals (Means, Small Sample Size)

• T-Interval: Used for estimating a population mean with a small sample size.
  • Inputs:
    • Sample mean (\bar{x}).
    • Sample standard deviation (s).
    • Sample size (n).
    • Confidence level.
  • Example: Mean = 21.6, Standard deviation = 2.3, Sample size = 18, Confidence = 0.95

Sample Standard Deviation Confidence Interval

• Formula:
  • \sqrt{\frac{(n-1)s^2}{\chi^21}} and \sqrt{\frac{(n-1)s^2}{\chi^22}}
  • n: Sample size.
  • s: Sample standard deviation.
  • \chi^21 and \chi^22: Chi-squared values from the Chi-squared distribution table.