Continuous Random Variables

📊 Introduction to Probability Distributions

Part 1 of 7 — The Normal Model

The Normal Distribution

The most important continuous probability distribution in statistics.

Key Properties:

• Bell-shaped and symmetric
• Mean = median = mode (at center)
• Described completely by $\\mu$ (mean) and $\\sigma$ (standard deviation)
• Notation: $X \\sim N(\\mu, \\sigma)$

The 68-95-99.7 Rule (Empirical Rule)

Example

Heights of adult women: $N(64, 2.5)$ inches

• 68% are between 61.5 and 66.5 inches
• 95% are between 59 and 69 inches
• 99.7% are between 56.5 and 71.5 inches

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Empirical Rule Practice 🧮

SAT scores follow $N(1060, 195)$.

1) 68% of scores fall between ___ and ___ (give the lower bound)

2) What’s the upper bound?

3) What percent of scores fall below $1060$?

📏 Z-Scores

Part 2 of 7 — Standardizing Values

The Z-Score Formula

$z = \\frac{x - \\mu}{\\sigma}$

A z-score tells you how many standard deviations a value is from the mean.

Standard Normal Distribution

When we standardize, we convert to $Z \\sim N(0, 1)$.

Example

Test scores: $\\mu = 75$, $\\sigma = 8$. A student scores 91.

$z = \\frac{91 - 75}{8} = \\frac{16}{8} = 2.0$

The student scored 2 standard deviations above the mean.

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Z-Score Calculations 🧮

Heights: $\\mu = 170$ cm, $\\sigma = 6$ cm.

1) Z-score for a height of 182 cm?

2) Z-score for a height of 158 cm?

3) A z-score of 0 corresponds to what height? (in cm)

📖 Normal Probabilities

Part 3 of 7 — Using the Standard Normal Table

Finding Probabilities

The Standard Normal Table (Table A) gives $P(Z < z)$ — the area to the LEFT of $z$.

Three Cases

Example

$P(Z < 1.25)$: Look up $z = 1.25$ in Table A → $0.8944$

$P(Z > 1.25) = 1 - 0.8944 = 0.1056$

$P(-1 < Z < 1.25) = P(Z < 1.25) - P(Z < -1) = 0.8944 - 0.1587 = 0.7357$

Working Backwards (Inverse Normal)

To find the z-score for a given percentile:

• 90th percentile: $z = 1.28$
• 95th percentile: $z = 1.645$
• 97.5th percentile: $z = 1.96$

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Normal Probability 🧮

$P(Z < 1.0) = 0.8413$ and $P(Z < -0.5) = 0.3085$.

1) $P(Z > 1.0) = ?$

2) $P(-0.5 < Z < 1.0) = ?$

3) What percent of a Normal distribution falls between $z = -0.5$ and $z = 1.0$?

🔍 Assessing Normality

Part 4 of 7 — Is the Data Normal?

Normal Probability Plot (Normal Quantile Plot)

A graph that plots each data value against its expected z-score if the data were perfectly Normal.

Interpretation:

• Roughly linear pattern → Data is approximately Normal
• Curved pattern → Data is NOT Normal
• S-shape → Data has outliers or is heavy-tailed

Other Methods to Assess Normality

When Can We Assume Normality?

• Large samples ($n \\geq 30$): CLT applies regardless
• Small samples: Need to check Normal probability plot
• Known Normal populations: Always OK

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Normality Assessment 🧮

1) A Normal probability plot shows an S-shaped curve. Is the data Normal? (yes/no)

2) For $n = 50$ data points, can you use Normal methods even if the population isn’t Normal? (yes/no)

3) What rule can you use to check if data roughly follows a Normal distribution? (state the name)

📊 Sampling Distributions

Part 5 of 7 — Sampling Distributions of $\\bar{x}$ and $\\hat{p}$

Sampling Distribution of $\\bar{x}$

If we take many samples of size $n$ from a population with mean $\\mu$ and SD $\\sigma$:

$\\bar{x} \\sim N\\left(\\mu, \\frac{\\sigma}{\\sqrt{n}}\\right)$

Sampling Distribution of $\\hat{p}$

For proportions from samples of size $n$ where population proportion is $p$:

$\\hat{p} \\sim N\\left(p, \\sqrt{\\frac{p(1-p)}{n}}\\right)$

Conditions: $np \\geq 10$ and $n(1-p) \\geq 10$

Key Insight

As $n$ increases, the standard deviation decreases — larger samples give more precise estimates.

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Sampling Distribution 🧮

Population: $\\mu = 500$, $\\sigma = 100$. Sample size $n = 25$.

1) Mean of the sampling distribution of $\\bar{x}$?

2) Standard deviation of $\\bar{x}$?

3) $P(\\bar{x} > 520) = P(Z > ?)$ — what is the z-score?

🏆 Problem-Solving Workshop

Part 6 of 7 — AP-Style Practice

Common AP Problem Types

1. Find a probability using z-scores and Table A
2. Find a percentile (inverse Normal)
3. Compare values from different distributions using z-scores
4. Sampling distribution questions about $\\bar{x}$ or $\\hat{p}$

Comparing Across Distributions

Who performed better?

• Student A: scored 680 on SAT $(\\mu = 500, \\sigma = 100)$
• Student B: scored 28 on ACT $(\\mu = 21, \\sigma = 5)$

$z_A = \\frac{680-500}{100} = 1.80 \\qquad z_B = \\frac{28-21}{5} = 1.40$

Student A has a higher z-score, so Student A did relatively better.

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Comparing Distributions 🧮

Math: $\\mu = 72, \\sigma = 10$. English: $\\mu = 80, \\sigma = 5$. A student scores 87 in Math and 88 in English.

1) Z-score in Math?

2) Z-score in English?

3) In which class did the student do relatively better? (math/english)

📝 Mixed Review

Part 7 of 7 — Comprehensive Review

Key Formulas

Checklist

• Normal distribution: bell-shaped, symmetric, $\\mu$ and $\\sigma$
• 68-95-99.7 Rule
• Z-scores and Table A
• Sampling distributions of $\\bar{x}$ and $\\hat{p}$
• Normal probability plots
• Central Limit Theorem

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Final Challenge 🧮

$X \\sim N(200, 25)$. A sample of $n = 100$ is taken.

1) $\\sigma_{\\bar{x}} = ?$

2) $P(\\bar{x} > 205)$: first find $z = ?$

3) Using Table A, $P(Z > 2) \\approx ?$