Question

Among a random sample of 500 college students, the mean number of hours worked per week at non-college-related jobs is 14.6. This mean lies 0.4 standard deviations below the mean of the sampling distribution. If a second sample of 500 students is selected, what is the probability that for the second sample, the mean number of hours worked will be less than 14.6?

Answer 1

The probability that for the second sample of 500 college students, the mean number of hours worked will be less than 14.6 is 0.6554

Explanation:

The sampling distribution of the sample mean is given by a normal distribution with mean
`\mu` and variance
`(\sigma^2)/(n)`, where
`\mu` is the mean and
`\sigma^2` is the variance of the population that generates the data. In this way the random variable;

`Z=(\bar x - \mu_(\bar x))/(\sigma_(\bar x))` is a standard normal variable. As
`\bar {x}-\mu_(\bar x) = 0.4\sigma_(\bar x)`, then
`Z = 0.4`.

`P (X <14.6) = P (Z <0.4) = 0.6554`