Question

The cost of treatment per patient for a certain medical problem was modeled by one insurance company as a normal random variable with mean $775 and a standard deviation of $150. What is the probability that the treatment cost of a patient is less than $1,000, based on this model? A. 0.8413 B. 0.1587 C. 0.8416 D. 0.1584

Answer 1

To solve this question, we will use the concept of z-scores and the properties of the normal distribution.

Step 1:
First, calculate the z-score. The z-score measures the number of standard deviations an element is from the mean. We find the z-score using the formula:

Z = (X - μ) / σ

Where:
- X is the value from the dataset (in this case, the cost of treatment we are interested in, $1000).
- μ is the mean of the dataset (in this case, $775).
- σ is the standard deviation of the dataset (in this case, $150).

Substitute the given values into the equation:

Z = (1000 - 775) / 150
Z = 1.5

So, the z-score is 1.5. This means that the cost of $1000 is 1.5 standard deviations above the mean.

Step 2:
Next, we will use the z-score to calculate the probability. We'll utilize the cumulative distribution function (CDF) of the normal distribution. The CDF at a point gives the probability that a random variable is less than or equal to that point.

The CDF for a z-score of 1.5 is around 0.93319.

Therefore, the probability that the treatment cost of a patient is less than $1000 is approximately 0.93319.

Looking at the provided options though, the closest one is option A: 0.8413