Find prob. of sample mean >= 50 from set of 4 distros

[mk]

If I want to find the probability of a sample mean of 50 or greater from a set of 4 distributions from 4 sample of 500 students, what are the correct steps?

My 4 samples are 40, 45, 42, & 53, and I know each of their Standard deviations (which are 8, 5, 15, & 4, respectively).

I think I need to convert each to a z score.

I believe I need to subtract average the 4 sample #'s, and then subtract that from 50.

Then Divide the standard deviation by the square root of 4?

Does this sound correct? I may be way off. I'm finding out that I'm not so great @ stats.
Thanks.

 

[royhaas]

You must compute the standard deviation of the combined sample means. Add the squares of the standard deviations and divide by the sample size. Then take the square root of the result, giving you the proper standard deviation. Use that standard deviation to compute the z-score. The numerator will be the sum of the deviations of the four sample means from the individual population means. The z-score corresponding to 50 will be (50-M)/S, where M is the sum of the population means, and S is the standard deviation of the combined sample means.

 

[mk]

Thanks, I truly appreciate your help! ( :