There is approximately a 94.32% probability that the duration of the new pain reliever for a random sample of 10 people is less than 260 minutes, based on a mean of 240 minutes and a standard deviation of 40 minutes.
To determine the probability of the drug's duration for a random sample of 10 people, we can use the normal distribution. Given a mean (μ) of 240 minutes and a standard deviation (σ) of 40 minutes, we can calculate the standard error of the mean (SEM) using the formula:
SEM = σ / sqrt(n)
where n is the sample size. For this scenario, n = 10.
SEM = 40 / sqrt(10) ≈ 12.65
Now, to find the probability, we convert the duration to a z-score using the formula:
z = (X - μ) / SEM
Assuming we want to find the probability of the drug lasting less than 260 minutes:
z = (260 - 240) / 12.65 ≈ 1.58
Consulting a standard normal distribution table or calculator, we find the corresponding probability. In this case, it's approximately 0.9432.
Therefore, there's about a 94.32% probability that the drug's duration for a random sample of 10 people is less than 260 minutes.