Final answer:
The random variable X represents the mean amount of coffee consumed per day by the 33 individuals surveyed. The distribution of X is a normal distribution with a mean of 16 ounces and a standard deviation of the population standard deviation (6.5 ounces) divided by the square root of the sample size (33), which is known as the standard error.
Step-by-step explanation:
The student asks about the distribution of the amount of coffee that people drink per day, which is normally distributed with a mean of 16 ounces and a standard deviation of 6.5 ounces, based on a sample of 33 individuals. This scenario is an application of the Central Limit Theorem, which states that the sampling distribution of the sample means approaches a normal distribution as the sample size gets larger, regardless of the population's distribution's shape, provided the sample is sufficiently large (usually n ≥ 30).
The random variable X, in this case, represents the mean amount of coffee consumed per day by the 33 individuals surveyed. Because we're considering the distribution of sample means, the random variable X should be written as Μx, where Μx is the sample mean. The assumed distribution of Μx will be approximately normal, thanks to the Central Limit Theorem, and will have a mean (μx) equal to the population mean (μ), which is 16 ounces, and a standard deviation (σx) equal to the population standard deviation (σ) divided by the square root of the sample size (n), hence σx = σ/√n = 6.5 ounces/√33.
The distribution of Μx is a normal distribution with the same mean as the population mean and its standard deviation is the population standard deviation divided by the square root of the sample size (known as the standard error). So, the distribution of Μx is N(μ = 16, σx = 6.5/√33).