Final answer:
To approximate the probability P(3.1 < X < 4.6) for a sample mean from an exponential distribution, the Central Limit Theorem is applied, and the standard normal distribution is used for calculation.
Step-by-step explanation:
The question asks to approximate the probability P(3.1 < X < 4.6) for a sample mean X of a random sample of size n = 48 from an exponential distribution with mean 4. To solve this, we apply the Central Limit Theorem, which states that the sampling distribution of the sample mean will be approximately normally distributed if the sample size is sufficiently large.
The mean (μ) of the sample mean distribution is the same as the mean of the population, in this case, 4. The standard deviation (σ) of the sample mean distribution is the population standard deviation divided by the square root of the sample size (σ/√n). In this case, the exponential distribution with mean 4 means the population standard deviation is also 4. So, the standard deviation of the sample mean is 4/√48. Using a standard normal distribution table or a calculator with normal distribution capabilities, convert the range to a Z-score and calculate P(3.1 < X < 4.6). The approximate probability can be found using a standard Z-table or software that computes areas under the normal curve.