Question

The time that a randomly selected individual waits for an elevator in an office building has a uniform distribution over the interval from 0 to 1 minute. For this distribution μ = 0.5 and σ = 0.289. (a) Let x be the sample mean waiting time for a random sample of 12 individuals. What are the mean and standard deviation of the sampling distribution of x? (Round your answers to three decimal places.)

Answer 1

The mean of the sampling distribution of x is 0.5 and the standard deviation is 0.083.

Explanation:

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For the population, we have that:

Mean = 0.5

Standard deviaiton = 0.289

Sample of 12

By the Central Limit Theorem

Mean = 0.5

Standard deviation
`s = (0.289)/(√(12)) = 0.083`

The mean of the sampling distribution of x is 0.5 and the standard deviation is 0.083.