The normal distribution is a fundamental continuous probability distribution used to model many natural phenomena such as IQ scores, height, and weight.
It differs from Year 1 distributions, which are discrete, while the normal distribution is continuous.
Characteristics
Parameters: Mean ((\mu)) and Variance ((\sigma^2)).
Symmetry: The distribution is symmetric (mean = median = mode).
Shape: Bell-shaped curve with asymptotes at each end.
Area: Total area under the curve equals 1.
Probability: (P(X = a) = 0) for any value (a), true for continuous distributions.
Notation
If a variable (X) follows a normal distribution with mean (\mu) and variance (\sigma^2), it is denoted as (X \sim N(\mu, \sigma^2)).
Sketching the normal curve is helpful when solving problems.
Finding Probabilities
Use the normal cumulative distribution function on calculators to find probabilities.
Example: To find (P(X < 33)) for (X \sim N(30, 2^2)), enter mean as 30, standard deviation as 2, upper bound as 33, and a very small value as the lower bound for accuracy.
Inverse Normal Distribution
Use the inverse normal function to find (x) such that (P(X < x) = p).
If calculators only return (P(X < x)), use (P(X > x) = 1 - P(X < x)) for problems requiring (P(X > x)).
Example: For (X \sim N(30, 2^2)), find (x) such that (P(X < x) = 0.4) gives (x = 29.5).
Approximating Binomial Distribution
The normal approximation is valid for binomial distribution when (n) is large and (p) is close to 0.5.
Parameters: (\mu = np), (\sigma^2 = np(1-p)).
Apply continuity correction ((\pm 0.5)) when using normal approximation for discrete distributions.
Example: For 600 bits with (p = 0.52), (X \sim N(312, 149.76)) and find (P(300 < X < 350)) using continuity correction.
Hypothesis Testing
Test hypotheses regarding the mean of a normal distribution using sample means.
Example: Test if the mean diameter of mats decreased from 9cm after servicing. Use (X \sim N(9, 0.15^2)) and sample mean distribution (\bar{X} \sim N(9, \frac{0.15^2}{30})).
Standard Normal Distribution
The standard normal distribution (Z \sim N(0, 1)) is used to standardize data.
Coding: (Z = \frac{X - \mu}{\sigma}).
Conversion allows easier probability calculations.
Example: For (X \sim N(50, 4^2)), convert (P(X > 55)) using standard normal.
Solving for Mean and Variance
Use standard normal distribution with given probabilities to find unknown parameters.
Example 5: Find (\mu) for (X \sim N(\mu, 5^2)) given (P(X < 18) = 0.9032).
Example 6: Find (\mu, \sigma^2) for (P(X < 17) = 0.8159) and (P(X < 25) = 0.9970) with two equations.
Continuity Correction Table
Adjustments for converting discrete to continuous distribution probabilities: