Probability and Normal Distribution: Study Notes for Statistics

Probability and the Normal Distribution

Probability of Random Variables

Probability is a measure of the likelihood that a particular event will occur. In statistics, random variables can take on different values, and we often want to find the probability that a variable falls within a certain range.

• Random Variable: A variable whose value is subject to variations due to chance.

• Probability: The proportion of times an event is expected to occur in the long run.

• Example: If a random variable X has a probability distribution, the probability that X is greater than 3 can be found by summing the probabilities of all values greater than 3.

Standard Normal Distribution (Z Distribution)

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. Probabilities for standard normal variables are found using Z-tables.

• Z-score: The number of standard deviations a data point is from the mean. Calculated as $Z = \frac{X - \mu}{\sigma}$.

• Finding Probabilities: Use Z-tables to find the probability that Z is less than or greater than a certain value.

• Example: Probability that Z < 1.13 is found by looking up 1.13 in the Z-table.

Table: Standard Normal Probabilities

Range	Probability
Z < 1.13	0.8708
-1.0 < Z < 0.36	0.6227
Additional info: Probabilities are found using cumulative Z-tables.

Normal Distribution and Applications

The normal distribution is a continuous probability distribution that is symmetric about the mean. Many natural phenomena follow a normal distribution.

• Mean ($\mu$): The average value.

• Standard Deviation ($\sigma$): A measure of spread or dispersion.

• Probability Calculation: To find the probability that X is between two values, convert both to Z-scores and use the Z-table.

• Example: For $\mu = 22.0$, $\sigma = 2.4$, probability that $19.7 < X < 25.3$:

$Z_1 = \frac{19.7 - 22.0}{2.4} = -0.958$ $Z_2 = \frac{25.3 - 22.0}{2.4} = 1.375$ Probability = P($Z_1 < Z < Z_2$)

Applications of the Normal Distribution

• Pregnancy Lengths: Mean = 268 days, $\sigma$ = 15 days. Probability that pregnancy lasts at least 300 days:

• Quartiles: The quartile $Q_2$ (median) of a normal distribution is equal to the mean.

• Volume of Soda Bottles: Mean = 32.3 oz, $\sigma$ = 1.2 oz. To find the percentage less than 32 oz, calculate Z and use the Z-table.

Table: Example Normal Distribution Applications

Scenario	Mean ($\mu$)	Std Dev ($\sigma$)	Calculation
Pregnancy > 300 days	268	15	$Z = \frac{300 - 268}{15}$
Height Quartile $Q_2$	63.6	2.5	$Q_2 = \mu$
Soda < 32 oz	32.3	1.2	$Z = \frac{32 - 32.3}{1.2}$

IQ Scores and Percentiles

Percentiles indicate the relative standing of a value within a data set. For normally distributed IQ scores, the cutoff for the top 14% can be found using the Z-table.

• Percentile: The value below which a given percentage of observations fall.

• Example: For mean = 100, $\sigma$ = 15, find the IQ score separating the top 14%:

Find Z such that P(Z > z) = 0.14. Use Z-table to find corresponding Z, then: $X = \mu + Z\sigma$

Sampling Distributions

Sampling distributions describe the distribution of a statistic (like the mean) over repeated samples from a population. As sample size increases, sample statistics tend to be closer to the population parameter.

• Sample Size Effect: Larger samples yield statistics with less variability and more accuracy.

• Key Point: Increasing sample size reduces the standard error, making sample statistics more reliable.

Table: Effect of Sample Size on Sampling Distribution

Sample Size	Variance	Accuracy
Small	High	Low
Large	Low	High

Additional info: All probability calculations involving the normal distribution require conversion to Z-scores and use of the standard normal table. Quartiles in a normal distribution are found using the mean and standard deviation. Sampling distributions are foundational for inferential statistics.