To solve this problem, we should use the concept of Z-scores, standard deviation, and the normal distribution.
Step 1: Identifying Given Information
Assume the mean cholesterol level in the given population is 180 dL. This is signified as the population mean, which is also statistically acknowledged as the expected value. Furthermore, the problem is looking for cholesterol levels that are 36 dL above the mean. The problem also tells us that the standard deviation of cholesterol levels in the population is 30 dL. The standard deviation is a measure of how dispersed the values are around the mean or expected value.
Step 2: Find the Z-Score
Initially, we need to calculate the Z-score, which measures how many standard deviations an element is from the mean. For cholesterol level 36 dL above the mean, we can calculate the Z-score by subtracting the population mean from the desired cholesterol level, and then dividing that by the population standard deviation. For this instance, the calculation will be:
(180 dL + 36 dL - 180 dL)/30 dL = 1.2
Hence, the cholesterol level of 216 dL is 1.2 standard deviations above the mean.
Step 3: Calculate the Probability
Finally, we will utilize the survival function, which is 1 minus the cumulative distribution function (CDF). The function calculates the probability that a variate assumes a value greater than the observed value. By inputting the Z-score into this function, we will gather the probability that a woman from the population has a cholesterol level greater than 216 dL. After performing this operation, we received the probability as 0.1150.
So, when we select a woman from the population at random, there is approximately an 11.51% chance that her cholesterol level will be more than 36dL above the mean.