Final answer:
The most likely value of x to occur in a sample from a population is the expected value or mean of the distribution, which can be estimated with the sample mean if the population parameters are unknown. The Central Limit Theorem allows us to use the sample mean as an approximation of the population mean for large enough samples.
Step-by-step explanation:
The value of x that is most likely to occur if we sample one observation at random from the population distribution is known as the expected value or mean of the distribution. When sampling from a population, if the distribution of the population is known, and if it is a normal distribution, the most likely value of x is the mean of the distribution. Often, a sample mean is used to estimate the population mean, especially when the population parameters are unknown. The Central Limit Theorem implies that the sampling distribution of the sample mean will be approximately normal, regardless of the shape of the original population distribution, if the sample size is sufficiently large.
In the provided examples, x represents a random variable from an unknown population. The distribution of random variable X needs to be identified to answer such probability questions. For instance, if the distribution is normal, the value that is most frequently occurring (the mean) can be calculated using the sample's statistics. When considering the empirical rule for bell-shaped distributions, roughly 95% of the samples will have the sample mean x within two standard deviations of the population mean μ.
In conclusion, the particular value of x most likely to occur will depend on the distribution of X. For example, if X follows a normal distribution with a known mean and standard deviation, that mean would be the most likely value to occur in a single random sample.