Normal Probability Distributions and Sampling Distributions: Study Notes

Normal Probability Distributions

Continuous Random Variables

A continuous random variable is a variable that can take infinitely many values, typically associated with measurements on a continuous scale without gaps or interruptions. Examples include height, weight, and temperature.

• Definition: A random variable is called continuous if its possible values form an interval of numbers.

• Example: If X is the height and Y is the weight of a randomly selected person, both X and Y are continuous random variables.

Density Function and Density Curve

Every continuous random variable has a density function. The graph of this function is called a density curve.

• Basic Properties:

  • The density curve is always on or above the horizontal axis.

  • The total area under the density curve (and above the horizontal axis) is 1.

• Interpretation: The area under the density curve within a specified range represents the percentage (probability) of all possible values of the variable that fall within that range.

Key Facts About Continuous Random Variables

• For any continuous random variable X and any real number c:

  • $P(X = c) = 0$

  • This means the probability that X takes any exact value is zero; only intervals have nonzero probability.

Uniform Random Variables (Uniform Distribution)

A random variable has a uniform distribution if its values are spread evenly over a range of possibilities. The density curve for a uniform distribution is rectangular.

• Density Function:

  • $f(X) = \begin{cases} \frac{1}{b-a} & \text{if } a \leq X \leq b \\ 0 & \text{otherwise} \end{cases}$

• Mean and Standard Deviation:

  • Mean: $\mu = \frac{a+b}{2}$

  • Standard deviation: $\sigma = \sqrt{\frac{(b-a)^2}{12}}$

• Example: For $a=1$, $b=4$, $P(2 < X < 3) = \frac{3-2}{4-1} = \frac{1}{3}$

Normal Random Variables (Normal Distribution)

A normal random variable is a continuous random variable with a density function that is symmetric and bell-shaped, characterized by two parameters: the mean ($\mu$) and the standard deviation ($\sigma$).

• Normal Curve: The graph of the normal density function. Different values of $\mu$ and $\sigma$ change the center and spread of the curve.

• Properties:

  • The mean, median, and mode of a normal distribution are all equal.

  • The curve is symmetric about the mean.

Standard Normal Distribution

The standard normal random variable (denoted Z) has mean $\mu = 0$ and standard deviation $\sigma = 1$.

• Standard Normal Curve: The total area under the curve is 1, and the curve extends indefinitely in both directions, approaching but never touching the horizontal axis.

• Symmetry: The curve is symmetric about 0.

• Probability Calculations: Probabilities are found using standard normal tables (Z-tables).

Empirical Rule (68-95-99.7 Rule)

The Empirical Rule describes the percentage of data within certain intervals in a normal distribution:

• Approximately 68% of data falls within 1 standard deviation of the mean.

• Approximately 95% falls within 2 standard deviations.

• Approximately 99.7% falls within 3 standard deviations.

Z-Scores and Percentiles

A z-score indicates how many standard deviations an observation is from the mean. Percentiles correspond to areas under the standard normal curve.

• Formula for z-score: $z = \frac{x - \mu}{\sigma}$

• Finding percentiles: Use the z-table to find the z-score corresponding to a given percentile, then convert to the original scale using $x = \mu + z\sigma$.

• Example: If IQs are normally distributed with $\mu = 100$, $\sigma = 16$, the 90th percentile is $x = 100 + 1.28 \times 16 = 120.48$.

Sampling Distributions

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean is the probability distribution of all possible sample means for samples of a given size from a population.

• Mean of the sample mean: $\mu_{\bar{x}} = \mu$ (the sample mean is an unbiased estimator of the population mean).

• Standard deviation of the sample mean: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

• Central Limit Theorem (CLT): For large sample sizes, the sampling distribution of the sample mean is approximately normal, regardless of the population's distribution.

• Example: If the population mean is 100 and standard deviation is 49, for $n=16$, $\sigma_{\bar{x}} = \frac{49}{4} = 12.25$

Sampling Distribution of the Sample Proportion

The sampling distribution of the sample proportion describes the probability distribution of sample proportions for samples of size n from a population.

• Population proportion: $p$ = proportion of population with a specified attribute.

• Sample proportion: $\hat{p} = \frac{x}{n}$, where $x$ is the number of successes in the sample.

• Mean of sample proportion: $\mu_{\hat{p}} = p$

• Standard deviation of sample proportion: $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$

• For large n: The sampling distribution of $\hat{p}$ is approximately normal.

Summary Table: Key Properties of Distributions

Distribution	Mean	Standard Deviation	Shape
Uniform	$\mu = \frac{a+b}{2}$	$\sigma = \sqrt{\frac{(b-a)^2}{12}}$	Rectangular
Normal	$\mu$	$\sigma$	Bell-shaped, symmetric
Sample Mean	$\mu_{\bar{x}} = \mu$	$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$	Approximately normal (for large n)
Sample Proportion	$\mu_{\hat{p}} = p$	$\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$	Approximately normal (for large n)

Key Terms

• Continuous random variable

• Density function

• Density curve

• Uniform distribution

• Normal distribution

• Standard normal distribution

• z-score

• Sampling distribution

• Central Limit Theorem

• Sample mean

• Sample proportion

• Unbiased estimator

Additional info: Some explanations and examples have been expanded for clarity and completeness, including formulas and context for sampling distributions and the Central Limit Theorem.