Finding the Parameters of a Normal Distribution
The core mechanism for finding unknown parameters is the standardization formula: . This formula relates a specific value from any normal distribution to its corresponding position on the standard normal curve.
In parameter-finding problems, the probability is typically known, but , , or both are unknown. The process requires working backward from the probability to find , and then solving for the missing variables.
This relationship can be rearranged into the linear form , which is often more convenient for algebraic manipulation, especially when dealing with simultaneous equations.
To find the -score, one must use the Inverse Normal Distribution method. This involves searching the body of a standard normal table for the given probability to find the corresponding value on the axes.
If the given probability is for a 'greater than' area (), it must first be converted to a 'less than' area () because standard tables typically provide cumulative area from the left.
High precision is required; students should use the critical values table for common probabilities (like 0.95 or 0.99) or use the 'add' columns in standard tables to find to three decimal places.
Step 1: Visualization. Sketch a normal curve and shade the area representing the given probability. This determines if the value is above or below the mean, which dictates the sign of the -score.
Step 2: Find z. Locate the -score in the table. If the shaded area is less than 0.5 and represents the left tail, the -score must be recorded as a negative value.
Step 3: Algebraic Solution. Substitute the known parameter (either or ), the value , and the -score into and solve for the remaining unknown.
When both and are unknown, the problem must provide two distinct probabilities for two different values of . This allows for the creation of a system of simultaneous equations.
Each probability is converted into its own -score ( and ). This results in two equations: and .
The most efficient way to solve these is usually by subtraction, which eliminates and allows for the calculation of . Once is found, it is substituted back into either equation to find .
| Scenario | Probability Condition | Position of | Sign of |
|---|---|---|---|
| Left Tail | Negative (-) | ||
| Right Tail | Positive (+) | ||
| Large Left Area | Positive (+) | ||
| Large Right Area | Negative (-) |
The Sketch Rule: Never attempt a parameter problem without a sketch. A visual representation prevents the common error of using a positive -score when the value is clearly below the mean.
Precision Check: Always use -scores to at least 3 decimal places. Rounding too early (e.g., to 2 decimal places) often leads to significant inaccuracies in the final values of and .
Sanity Testing: Once and are calculated, verify they are logical. For instance, must always be positive, and should lie between the two values if one is positive and the other is negative.