Suppose you draw a value each day at random from a normal distribution with a mean of 0 and a standard deviation of How many days should it take on average for you to draw a value that's greater than 2?

Question Analysis

The question asks you to calculate the expected number of days it will take to draw a value greater than 2 from a normal distribution with a mean of 0 and an unspecified standard deviation. Since the standard deviation is not provided, we assume it to be 1 for a standard normal distribution. The problem involves understanding the properties of the normal distribution and using probabilities to find the expected number of trials (days) needed to achieve a certain outcome (drawing a value greater than 2).

Answer

To solve this problem, we need to determine the probability of drawing a value greater than 2 from a standard normal distribution, which has a mean (μ) of 0 and a standard deviation (σ) of 1.

1. Calculate the Probability:

   • Use the standard normal distribution table (Z-table) or a statistical tool to find the probability (P) that a draw is greater than 2.
   • The Z-score for a value of 2 is calculated as follows:
     [
     Z = \frac{X - \mu}{\sigma} = \frac{2 - 0}{1} = 2
     ]
   • From the Z-table, P(Z > 2) ≈ 0.0228. This means there is approximately a 2.28% chance of drawing a value greater than 2 on any single day.

2. Calculate the Expected Number of Days:

   • The expected number of days ( E ) to draw a value greater than 2 is the reciprocal of the probability:
     [
     E = \frac{1}{P(Z > 2)} = \frac{1}{0.0228} \approx 43.86
     ]
   • Therefore, on average, it should take approximately 44 days to draw a value greater than 2.

In summary, by understanding the properties of the normal distribution and calculating the probability of drawing a value greater than 2, you can determine that it takes about 44 days on average to achieve this outcome.