Question

Unusual events: Events are unusual if their probability is 5% (0.05) or less. When asked about unusual events as part of a probability distribution, you have to do the following: 1) Draw the normal distribution with appropriate mean and st dev.

Answer 1

You can plot the normal distribution curve with μ = 50 and σ = 10 and shade the area to the left of Z = -1.645 to represent the unusual event.

To represent an event as unusual based on a probability distribution, you can follow these steps:

Determine the mean (average) and standard deviation (SD) of the distribution. These parameters are essential for creating a normal distribution curve.

Calculate the Z-score for the probability of interest (in this case, 0.05 or 5%). The Z-score measures how many standard deviations a particular data point is from the mean. You can use the formula:

Z = (X - μ) / σ

Where:

Z is the Z-score

X is the value for which you want to find the probability

μ is the mean of the distribution

σ is the standard deviation of the distribution

Use the Z-score to find the corresponding value on the standard normal distribution table (or use a calculator or software that can provide this information).

Plot the normal distribution curve with the calculated mean and standard deviation, and shade the area representing the unusual event (in this case, the area to the left of the Z-score).

Let's assume a hypothetical example:

Mean (μ): 50

Standard Deviation (σ): 10

Probability for unusual event (p): 0.05

First, calculate the Z-score:

Z = (X - μ) / σ

Z = (X - 50) / 10

Now, set Z equal to the Z-score corresponding to a 5% probability (from a standard normal distribution table or calculator):

Z ≈ -1.645

The shaded area to the left of Z = -1.645 represents the unusual event with a probability of 5% or less.