Final answer:
The question involves finding the distribution of the sample mean and probabilities for sample observations. It requires using the Central Limit Theorem for large samples and appropriate formulas for exponential and uniform distributions. Probabilities are calculated over intervals for continuous variables, as individual values in such distributions have a probability of zero.
Step-by-step explanation:
The question requires knowledge of basic statistical concepts such as means, variances, and probability distributions. Specifically, the student is asked to deal with normal and exponential distributions, probabilities for continuous random variables, and sampling distributions of sample means.
Typically, if x has a distribution with a mean (μ) of 18 and a standard deviation (σ) of 12, then for a sufficiently large sample size, by the Central Limit Theorem, the distribution of the sample mean &xmacr; will be approximately normal with mean μ (&xmacr;) = 18 and standard deviation σ (&xmacr;) = σ/√n.
To calculate the probability P(18 ≤ &xmacr; ≤ 20) for a sample of size 38, one could use the Z-score formula and look up the values in the standard normal distribution tables. The same process applies for a sample of size 68.
Probability calculations for individual observations, however, use the specified distribution of the variable. If X follows an exponential distribution with λ = 1/12, P(x > 20) would be calculated using the exponential formula P(X > x) = e-(λx).
Distribution of Sample Mean and Probabilities
For a uniform distribution X ~ U(0, 20), P(2 < x < 18) can be found by recognizing the uniform distribution's constant probability density function. As X is continuous, P(x = c) is always 0. Instead, we look for probabilities over an interval (c, d), such as P(x > 5) being 1 minus the probability of x being less than or equal to 5.
When considering the normal distribution X~ N(60, 9) and forming random samples of 25, the random variable for the sample mean &xmacr; would have its own distribution N(μ, σ2/n). One can then find specific probabilities and percentiles for the distribution of &xmacr;.