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 (at the 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

Concept Check 🎯

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.

Concept Check 🎯

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: $z = 1.96$

Concept Check 🎯

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?

Look at a Graph of the Data

🔑 Real data are never exactly Normal. The question is whether a Normal model is a reasonable approximation.

When Can We Use Normal Methods?

• Data: a Normal model fits the data only if the graph looks roughly symmetric and unimodal with no outliers.
• Sample means: the sampling distribution of $\bar{x}$ is approximately Normal if the population is Normal or $n \geq 30$ (Central Limit Theorem).
• Small samples: check a graph of the sample for strong skewness and outliers.

Concept Check 🎯

Normality Assessment 🧮

1) A histogram of the data shows two distinct peaks. Is a Normal model reasonable? (yes/no)

2) For $n = 50$ observations from a skewed population, is the sampling distribution of $\bar{x}$ approximately Normal? (yes/no)

3) What rule compares the percent of data within 1, 2 and 3 SDs to a Normal model? (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.

Concept Check 🎯

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.

Concept Check 🎯

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 $\hat{p}$
• Checking normality with graphs and the 68-95-99.7 rule
• Central Limit Theorem

Concept Check 🎯

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, $P(Z > 2) \approx ?$