Probability -- Normal Distributions and Binomial Experiments

Binomial Experiment

An experiment with two possible outcomes or a success/fail outcomes.

Conditions for a Binomial Setting

BINS

• Binary success/fail outcome,

• Independent trials,

• Number of n trials fixed before experimentation,

• Success probability fixed.

Binomial Formula

nCr*(p(s))^r(1-p(s))^n-r

Definitions in the Binomial Formula

• n=number of trials,

• r=number of successes,

• p(s)=probability of success

Expected Value of a Binomial Distribution

E(x)=n*p(s)

When are Normal Distributions often used?

For standardizing data sets, ex. SAT scores, heights, travel times.

Normal Distribution Conditions

• Single-peaked

• Roughly symmetric (about the mean)

• Mean is approximate to the Median

• No obvious outliers

• No significant gaps in data

• Roughly Bell shaped

Empirical Rule of Normal Distributions

• Around 68% of outcomes fall within the standard deviation of the mean.

• Around 95% of outcomes fall within two standard deviations of the mean.

• Around 99.7% of outcomes fall within three standard deviations of the mean.

(Optional) How would we find the percent of data under the curve, or the area under the curve, if the standard deviation is a decimal value?

We would use the normal distribution formula on a graphing calculator and use an integral function to add up each individual rectangle’s area under the portion of the curve within the standard deviation.

Z-score Formula

Z=(x-mu)/sigma

What is it called when we standardize data?

We assign a z-score to the outcome, assuming the data set is normally distributed.