Normal Probability Distributions and the Central Limit Theorem Chapter. 5

Normal Probability Distributions

Introduction to Normal Distributions and Standard Normal Distributions

Normal probability distributions are fundamental in statistics for modeling continuous random variables. The normal distribution, also known as the Gaussian distribution, is characterized by its bell-shaped, symmetric curve and is widely used due to the Central Limit Theorem.

• Continuous random variable: A variable that can take any value within a given interval (e.g., hours spent studying in a day: 0.0 to 24.0).

• Continuous probability distribution: Describes probabilities for continuous random variables using a probability density function (PDF).

• Normal distribution: The most important continuous probability distribution in statistics. The total area under the curve equals 1.00 (100%).

Properties of a Normal Distribution

The normal distribution has several defining properties that make it useful for statistical analysis.

• Mean, median, and mode are equal (or approximately equal).

• The curve is bell-shaped and symmetric about the mean.

• Total area under the curve is 1.00.

• The curve approaches but never touches the x-axis as it extends further from the mean.

• Inflection points occur at $\mu - \sigma$ and $\mu + \sigma$ (one standard deviation below and above the mean).

Empirical Rule (68-95-99.7 Rule):

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

• Approximately 95% within 2 standard deviations.

• Approximately 99.7% within 3 standard deviations.

Interpreting Normal Distributions

Normal distributions are used to model real-world data, such as standardized test scores. The mean and standard deviation can be estimated using the inflection points of the curve.

• Example: If the mean test score is 18.0 and the inflection points are at 12.2 and 23.8, the standard deviation is approximately $(23.8 - 12.2)/2 = 5.8$.

Standard Normal Distribution and Z-Scores

Definition and Transformation

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. Any normal distribution can be converted to the standard normal distribution using z-scores.

• Z-score formula:

$z = \frac{\text{Value} - \text{Mean}}{\text{Standard deviation}}$

• Z-scores indicate how many standard deviations a value is from the mean.

• Allows comparison across different normal distributions.

Using the Standard Normal Table

The standard normal table (z-table) provides the cumulative area (probability) to the left of a given z-score.

• To find the area to the right of a z-score, subtract the table value from 1.

• To find the area between two z-scores, subtract the smaller area from the larger area.

Example: To find the probability that $z$ is between -0.75 and 1.23:

• Area to left of 1.23: 0.8907

• Area to left of -0.75: 0.2266

• Probability: $0.8907 - 0.2266 = 0.6641$

Converting Between X Values and Z-Scores

• To convert an x-value to a z-score: $z = \frac{x - \mu}{\sigma}$

• To convert a z-score to an x-value: $x = \mu + z\sigma$

Example: If $\mu = 9$, $\sigma = 2$, and $z = 1.96$, then $x = 9 + 1.96 \times 2 = 12.92$.

Sampling Distributions and the Central Limit Theorem

Sampling Distributions

A sampling distribution is the probability distribution of a sample statistic (such as the mean) based on all possible samples of a given size from a population.

• The mean of the sampling distribution of the sample mean ($\mu_{\bar{x}}$) equals the population mean ($\mu$).

• The standard deviation of the sampling distribution (standard error) is $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$.

Central Limit Theorem (CLT)

The Central Limit Theorem states that, for sufficiently large sample sizes ($n \geq 30$), the sampling distribution of the sample mean will be approximately normal, regardless of the population's distribution.

• Mean of sampling distribution: $\mu_{\bar{x}} = \mu$

• Variance: $\sigma^2_{\bar{x}} = \frac{\sigma^2}{n}$

• Standard error: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

Example: If the mean sleep time is 6.9 hours with SD 1.5 hours, and samples of 100 students are taken:

• Mean of sampling distribution: $6.9$

• Standard error: $\frac{1.5}{\sqrt{100}} = 0.15$

Probabilities for Sampling Distributions

To find probabilities involving sample means, use the standard error in the z-score formula:

$z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$

• Example: If the mean room and board expense is $11,806 with SD $1,650, and a sample of 9 colleges is taken, the standard error is$\frac{1650}{\sqrt{9}} = 550$.

Normal Approximations to Binomial Distributions

Conditions for Normal Approximation

The normal distribution can approximate the binomial distribution when both $np > 5$ and $nq > 5$ (where $q = 1 - p$).

• Mean: $\mu = np$

• Standard deviation: $\sigma = \sqrt{npq}$

Continuity Correction

Since the binomial distribution is discrete and the normal is continuous, a continuity correction of 0.5 is applied when approximating binomial probabilities with the normal distribution.

• For $P(x = c)$, use $P(c - 0.5 < x < c + 0.5)$

• For $P(x < c)$, use $P(x < c - 0.5)$

• For $P(x > c)$, use $P(x > c + 0.5)$

Original Binomial Probability	Normal Approximation (with Continuity Correction)
P(x = c)	P(c - 0.5 < x < c + 0.5)
P(x < c)	P(x < c - 0.5)
P(x > c)	P(x > c + 0.5)
P(x ≤ c)	P(x < c + 0.5)
P(x ≥ c)	P(x > c - 0.5)

Example: Normal Approximation to Binomial

• Suppose $n = 45$, $p = 0.40$, $q = 0.60$.

• Mean: $\mu = 45 \times 0.40 = 18$

• Standard deviation: $\sigma = \sqrt{45 \times 0.40 \times 0.60} \approx 3.2$

• To find $P(x < 20)$, use $P(x < 19.5)$ with the normal distribution.

Summary Table: Key Formulas

Concept	Formula
Z-score (single value)	$z = \frac{x - \mu}{\sigma}$
Z-score (sample mean)	$z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$
Mean of binomial	$\mu = np$
SD of binomial	$\sigma = \sqrt{npq}$
Standard error (sample mean)	$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

Additional info:

• These notes are based on lecture slides for a college-level statistics course, focusing on normal probability distributions, the standard normal distribution, sampling distributions, the Central Limit Theorem, and normal approximations to the binomial distribution.

• All examples and formulas are standard in introductory statistics textbooks.