Normal Distribution Intro

Probability Distributions

This section review the concept of a random variable and probability distribution.

Random Variable

Many real-life decisions are made by assigning probabilities to all possible outcomes of a particular situation. For example, a bank's lender specializes in obtaining loan approvals for start-ups. The lender can compute the probability that they will get zero, one, two, etc loans a month.

Continuous random variables cannot be counted, but they can be measured. A continuous random variable has an uncountable number of possible outcomes, which is represented by an interval on the real number line.

• can take on an infinite number of values

  • decimal

  • fraction

Examples include: Weight, height, time, etc.

Continuous Random Variable

A random variable is discrete if it has a finite or countable number of possible outcomes that can be listed.

Discrete Random Variable

Discrete vs Continuous

​Discrete vs Continuous Examples

Probability Distributions

Summary

The distribution of a random variable tells which values a random variable can take on and how often it takes on those values. In doing so, a probability distribution describes a population because when items are sampled from a population, the value observed is the value of a random variable.

-Can be represented as graphs, tables, and formulas

-Can be discrete or continuous

Multiple Choice

The center of the curve represents what value of a data set?

• μ = population mean

• δ = population standard deviation

• Σ= sum of the data values

• Ζ = z or standard scores

Multiple Choice

What is Standard Deviation?

• A measure of central tendency

• A measurement of how much data varies

• A measurement of how much data varies from the average

Standard Deviation

The standard deviation on a bell curve is what the x axis is going up and down by. On the graph given, the standard deviation is 5 because the graph is going up and down by 5.

​Empirical Rule / Standard Normal Distribution

Multiple Choice

What percentages of pregnancies last less than 250 days?

• 2.35%

• 13.5%

• 16%

• 34%

Multiple Choice

What percentage of pregnancies are longer than 298 days?

• 0.15%

• 2.35%

• 2.5%

• 13.5%

Multiple Choice

If a pregnancy lasts 282 days, in what percentile is it? (Percentile is a fancy way of saying “the percent below this point.”)

• 16th

• 50th

• 84th

• 95th

Multiple Choice

A grading scale is set up for 1000 students’ test scores. It is assumed that the scores are normally distributed with a mean score of 75 and a standard deviation of 15. Calculate the score that lies -2 standard deviation from the mean.

• 60

• 475

• 45

• 9500

Multiple Choice

What percent of a normal distribution lies outside of ±3 standard deviations?

• 99.7%

• .15%

• .3%

• 100%

Multiple Choice

In an Oreo factory, the mean mass of a cookie is given as 40 g. For quality control, the standard deviation is 2 g. If 10 000 cookies were produced, how many cookies are within 2 g of the mean?

• 6800

• 9500

• 3200

• 9750