Question

Let x1, x2, x3, x4 be independent random variables each having the same distribution with mean 5 and standard deviation 3. a) True

b) False

Answer 1

The question pertains to uniform distribution, sampling distribution, and properties of sums of random variables, with a focus on the central limit theorem and law of large numbers.

Step-by-step explanation:

The question centers around the properties of uniform distribution and properties of distributions of sums of random variables. For a set of independent random variables x1, x2, x3, and x4 with the same distribution, mean, and standard deviation, certain statistical rules apply. These rules are related to the central limit theorem which describes the characteristics of the sampling distribution of a sample mean.

A uniform distribution X U(a, b) implies that each outcome within the range a to b is equally likely. The mean (µ) of a uniform distribution can be calculated using the formula µ = (a+b)/2, and the standard deviation is σ = (b-a)/√12.

Key True/False Statements:

• True: When the sample size is large, the mean of the sampling distribution of the means is approximately the same as the mean of the population (law of large numbers).
• False: When the sample size is large, the standard deviation of the sampling distribution of the means is not the same as the standard deviation of the population since it decreases by the square root of the sample size (standard error).