Final answer:
The probability that the medication will take more than 20.5 minutes to begin reducing symptoms is found by calculating the z-score and using the standard normal distribution table. The correct answer is option B, 0.0301.
Step-by-step explanation:
The question asks to determine the probability that it will take more than 20.5 minutes for a certain pain reliever to begin reducing symptoms, given that the mean time is 15 minutes with a standard deviation of 2.92 minutes, assuming a normal distribution.
To find this probability, we will calculate the z-score for the time of 20.5 minutes:
Z = (X - μ) / σ
Where:
X = the value we are looking at, which is 20.5 minutes
μ = the mean, which is 15 minutes
σ = the standard deviation, which is 2.92 minutes
Now we plug in the values:
Z = (20.5 - 15) / 2.92
Z = 5.5 / 2.92
Z ≈ 1.88
Next, we look up the z-score of 1.88 in the standard normal distribution table to find the probability of being below this value, which is typically around 0.9699. To find the probability of being above this z-score, we subtract this from 1:
P(X > 20.5) = 1 - P(Z < 1.88)
P(X > 20.5) = 1 - 0.9699
P(X > 20.5) = 0.0301
Therefore, option B, 0.0301, is the correct probability that the medication will take more than 20.5 minutes to begin reducing symptoms.