# probability and random sample

suppose that a body mass index for a population of 30-60 year old men follows a normal distribution with mean 26 and standard deviation 4. If we take a random sample of 7 men age 30-60 years old. whe is the probability that the average BMI of these 7 men is equal to or larger than 30?

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The sample average BMI $\bar{X}$ of the $7$ people has normal distribution, mean $26$, standard deviation $\frac{4}{\sqrt{7}}$.
We want the probability that $\bar{X}\ge 30$. This is the probability that a standard normal $Z$ is $\ge \frac{30-26}{4/sqrt{7}}$. This probability can be cound using tables of the standard normal, or software. Actually, most software will evaluate the probability that $\bar{X}\ge 30$ directly, if you enter the population mean and standard deviation, and the sample size.