Module 13 Wrap Up: Probability Distribution for Normal Continuous Quantitative Variables

Continuous Random Variables

• Continuous random variables are not limited to distinct values (e.g., foot length, weight).
• Cannot display the probability distribution for these variables using tables or histograms.
• Use a probability density curve to assign probabilities to intervals of x values.
• Probabilities are found by calculating the area under the curve.

Normal Probability Density Curve

• Used to model probability distribution for variables such as weight, shoe sizes, and foot lengths.
• Normal curves are mathematical models:
  • μ (mu) represents the mean.
  • σ (sigma) represents the standard deviation.
• Greek letters indicate that the normal curve is a mathematical model, not a real data distribution.
• Represents a perfect bell-shaped distribution.

Empirical Rule

• 68% of observations fall within one standard deviation of the mean.
• 95% within two standard deviations.
• 99.7% within three standard deviations.

Z-scores

• Used to standardize values from different distributions.
• Measures how far x is from the mean in standard deviations.
• Z-score interpretation:
  • A z-score of 1 means the x value is one standard deviation above the mean.

Standard Normal Distribution

• Distribution of z-scores is also a normal probability density curve.
• This curve is called the standard normal distribution.
• Use an applet with the standard normal curve to find probabilities for any normal distribution.

Working with Probabilities and X Values

• Can work backwards to find the x value for a given probability.
• Use different outputs to work backwards from probabilities to x values.
• Applet used to find x values corresponding to quartiles and percentiles.