Question

Suppose the population is all VCU students who track their steps each day. Among these students suppose the mean number of steps per day is 8989 with a standard deviation of 688. There are a few students who take many steps per day, and hence the distribution is skewed heavily to the right. If a simple random sample of 169 VCU students who track their steps each day is selected and the number of steps per day is determined for each, describe completely the sampling distribution of , the resulting mean number of steps per day for this sample of 169 VCU students who track their steps each day.

Answer 1

By the Central Limit Theorem, the sampling distribution is approximately normal, with mean 8989 and standard deviation 52.92.

Explanation:

Central Limit Theorem

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.

Mean number of steps per day is 8989 with a standard deviation of 688.

This means that
`\mu = 8989, \sigma = 688`

Describe completely the sampling distribution of the resulting mean number of steps per day for this sample of 169 VCU students who track their steps each day.

Sample of 169 means that
`n = 169`

By the Central Limit Theorem, the sampling distribution is approximately normal, with mean
`\mu = 8989` and standard deviation
`s = (688)/(√(169)) = 52.92`.