Normal Probability Distributions and Sampling Distributions: Study Notes

Normal Probability Distributions

Density Curve

A density curve is a graphical representation of a continuous probability distribution. It shows the relative likelihood of different outcomes for a random variable.

• Properties:

  • The area under the curve equals 1.

  • Every value on the curve must be non-negative (≥ 0).

• Area under the curve: Corresponds to probability.

Uniform Distribution

The uniform distribution is a type of continuous probability distribution where all outcomes in a given interval are equally likely.

• Definition: All values within a specified range have the same probability.

• Graph: The graph is a rectangle over the interval.

• Formula: For a uniform distribution over [a, b], the probability density function (pdf) is: $f(x) = \frac{1}{b-a}$ for $a \leq x \leq b$

• Example: If voltage is uniformly distributed between 13.5 and 15.5 volts, the probability that a randomly selected voltage is greater than 13.75 volts is calculated by finding the area under the curve from 13.75 to 15.5.

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution with a mean of 0 and a standard deviation of 1.

• Properties:

  • The graph is bell-shaped and symmetric.

  • Mean ($\mu$) = 0

  • Standard deviation ($\sigma$) = 1

  • The total area under the curve is 1.

• Probability: The area under the curve between two points corresponds to the probability that a value falls within that interval.

z-Scores

A z-score measures the number of standard deviations a data point is from the mean. It is used to standardize values and compare them across different normal distributions.

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

  • $x$ = observed value

  • $\mu$ = mean

  • $\sigma$ = standard deviation

• Interpretation: Positive z-scores indicate values above the mean; negative z-scores indicate values below the mean.

z-Scores vs. Areas

z-scores are used to locate positions on the standard normal curve, while areas under the curve represent probabilities.

• Area: The area between two z-scores corresponds to the probability that a value falls between those scores.

• Calculation: Use standard normal tables or technology to find areas.

Finding Probabilities Given z-Scores

To find the probability that a value falls within a certain range, use the z-score and standard normal table.

• Examples:

  • $P(a < z < b)$: Probability that z is between a and b.

  • $P(z < a)$: Probability that z is less than a.

  • $P(z > a)$: Probability that z is greater than a.

  • $P(a < z < b) = P(z < b) - P(z < a)$

• Visualization: Shade the area under the curve between the relevant z-scores.

Finding z-Scores Given Probabilities

To find the z-score corresponding to a given probability (percentile), use the standard normal table in reverse.

• Procedure:

  1. Draw the standard normal curve and shade the region corresponding to the given probability.

  2. Use the table or technology to find the z-score that matches the area.

• Example: Find the z-score for the 90th percentile.

Nonstandard Normal Distributions

When the normal distribution has a mean and standard deviation other than 0 and 1, it is called a nonstandard normal distribution.

• Procedure:

  1. Convert the value to a z-score using $z = \frac{x - \mu}{\sigma}$.

  2. Use the standard normal table to find probabilities or areas.

• Applications: Used to find probabilities and percentiles for real-world data.

• Example: If exam scores are normally distributed with mean 70 and standard deviation 8, find the probability that a student scored at least 88.

Finding Probabilities and Values: Examples

• Finding Probabilities: Given a normal distribution, calculate the probability that a value falls above, below, or between certain values.

• Finding Values: Given a percentile, find the corresponding value in the original distribution.

• Example: If only the tallest 5% of men are eligible for a program, find the minimum height requirement.

Sampling Distributions

Sampling Distribution of a Statistic

The sampling distribution of a statistic is the probability distribution of that statistic for all possible samples of a given size from the same population.

• Common statistics: Mean, variance, proportion.

• Probability distribution: Can be described by a table, formula, or probability histogram.

• Example: The distribution of sample means for all samples of size n from a population.

Sampling Distribution of the Sample Mean ($\bar{x}$)

The sampling distribution of the sample mean describes the distribution of means from all possible samples of a given size.

• Central Limit Theorem (CLT): For a population with any distribution, the distribution of the sample mean approaches a normal distribution as the sample size increases.

• Conditions:

  • If the original population is normally distributed, the sample mean is normally distributed for any sample size.

  • If the original population is not normally distributed, the sample mean is approximately normal when $n > 30$.

• Mean and Standard Deviation:

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

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

• Two distributions: One for the population, one for the sample mean.

When to Use Which Distribution?

• Individual Value: Use normal distribution ideas for single values.

• Sample Mean: Use sampling distribution formulas for means from samples.

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

Central Limit Theorem: Examples

• Example 1: SAT scores are normally distributed. Find the probability that the mean score for a sample is between two values.

• Example 2: Heights of men are normally distributed. Find the probability that the mean height of a sample is less than a certain value.

Summary Table: Key Distributions and Formulas

Distribution	Mean	Standard Deviation	Formula
Standard Normal	0	1	$z = \frac{x - \mu}{\sigma}$
Nonstandard Normal	$\mu$	$\sigma$	$z = \frac{x - \mu}{\sigma}$
Sampling Distribution of $\bar{x}$	$\mu$	$\frac{\sigma}{\sqrt{n}}$	$z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$
Uniform	$\frac{a+b}{2}$	$\frac{b-a}{\sqrt{12}}$	$f(x) = \frac{1}{b-a}$ for $a \leq x \leq b$

Additional info:

• Some slides referenced the use of technology (e.g., Statdisk.com) for finding probabilities and z-scores.

• Extra credit opportunities were mentioned, but not included in the academic notes.