Answer: To find the probability that a person chosen at random has a median lifetime income between 1 to 2 standard deviations below the mean, we need to calculate the area under the normal distribution curve within that range.
First, let's define the variables:
μ = Mean lifetime income = $25800
σ = Standard deviation = $14000
We want to find the probability of having a median lifetime income between 1 to 2 standard deviations below the mean.
1 standard deviation below the mean would be μ - σ, and 2 standard deviations below the mean would be μ - 2σ.
μ - σ = $25800 - $14000 = $11800
μ - 2σ = $25800 - 2 * $14000 = $-2200
Next, we need to find the z-scores for these values. The z-score represents the number of standard deviations a given value is from the mean in a standard normal distribution.
For μ - σ:
z1 = (11800 - μ) / σ = (11800 - 25800) / 14000 ≈ -1.5714
For μ - 2σ:
z2 = (-2200 - μ) / σ = (-2200 - 25800) / 14000 ≈ -2.2857
Using a standard normal distribution table or a statistical software, we can find the corresponding probabilities associated with these z-scores.
The probability of having a median lifetime income between 1 to 2 standard deviations below the mean is the difference between the probabilities corresponding to z1 and z2.
P(1 to 2 standard deviations below the mean) = P(z1 < Z < z2)
You can refer to a standard normal distribution table or use statistical software (such as Excel, R, or Python) to calculate the probabilities. The exact values may vary depending on the specific table or software used.