# Find a Probability of a Normally Distributed Random Sample

Please help me figure out how to do this problem. I need to be able to understand how to solve problems like this. Thanks times a million!

Problem: An employer is interested in the commute times for its employees. The commute times (traveling from home to work) of all 2500 employees were recorded. The mean is 23.1928 minutes and the standard deviation is 9.438181 minutes. In a random sample of 10 people, what's the probability that the sample mean commute time will be less than 15 minutes?

note: The data is normally distributed.

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Let $\bar{X}$ be the sample mean of the commute time of $10$ randomly chosen employees. The mean of the random variable $\bar{X}$ is the population mean. We can assume that this is $23.1928$. The standard deviation $\tau$ of $\bar{X}$ is given by $\tau=\frac{\sigma}{\sqrt{10}}$, where $\sigma$ is the population variance, which we can assume is $9.438181$.

Calculate $\tau$. The probability that $\bar{X}$ is $\le 15$ is the probability that a standard normal $Z$ is $\le \frac{15-\mu}{\tau}$, where $\mu$ is the mean (known) and $\tau$ has just been calculated.

After you have calculated $\frac{15-\mu}{\tau}$, you can find the answer in a table of the standard normal. Not quite, since $\frac{15-\mu}{\tau}$ is negative. But the probability that $Z\le -a$ is the same as the probability that $Z\ge a$. For positive $a$, you can (almost) find this information in the usual tables.

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Thank you so much! So the fact that N=2500 and n=10, with N>>n, means I can assume the population mean is the same (well, approximate) for the sample mean. Thanks again! –  Becky Mar 4 '13 at 2:59
Yes, $2500$ is huge. –  André Nicolas Mar 4 '13 at 3:12
If it's not too much to ask, I'd like one last piece of advice. I've done several more problems like this now, and they've all been correct. But a new problem (same N=2500, population mean, and population standard deviation) is asking, in a random sample of 15, what's the probability that the sample mean commute time will be between 18 and 42 minutes? I've done calculations (new standard dev=2.436928) to the point of getting z values: lower z=-2.13 and upper z=7.7. Surely I must've done something wrong, because my class only has a z table going up to a z value of 3.4! What's with the 7.7? –  Becky Mar 4 '13 at 3:38
Looks reasonable, here the standard deviation is smallish, under $2.5$. And $42$ is way out. For all practical statistical purposes, $Z$ is "never" $\gt 4$. The probability you are looking for is almost exactly equal, to many many decimal places, to the probability we are $\gt -2.13$. The reason the table runs out is that there is no point in extending the table and printing $1.0000, 1.0000,1.0000,\dots$. –  André Nicolas Mar 4 '13 at 3:50
Thanks! You've helped me a bunch! –  Becky Mar 4 '13 at 17:21