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Population proportion, given mean and standard deviation?

How do I go about calculating these?

1:

"A normal population has a mean of 34 and a standard deviation of 11. What proportion of the population is less than 41?"

I'm confused because there's no sample size given.

and 2:

"A bottler of drinking water fills plastic bottles with a mean volume of 999 mL and standard deviation 4 mL. The fill volumes are normally distributed. What proportion of bottles have volumes greater than 994 mL?"

Thanks

1 Answer

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  • 5 years ago
    Favorite Answer

    Given that this is a normal population, and you know the mean and standard deviation, we can solve this using z-scores.

    The z-score of a given value is the number of standard deviations away from the mean a value is. So using this equation:

    n = m + sz

    We can find the z-score.

    We're looking for the value 41 (n) with mean 34 (m) and standard deviation 11 (s). Plug in what we know and solve for the unknown:

    41 = 34 + 11z

    41 = 34 + 11z

    7 = 11z

    z = 7/11 ≈ 0.636

    From here, we can use the z-score against a z-score table to give us the probability that a random data point will be less than the point in question. That's exactly what we're being asked (what proportion of the population is less than 41), so that is the answer to our question.

    I use the link below for z-score tables:

    http://www.regentsprep.org/regents/math/algtrig/at...

    From there, look for z = 0.636 and that gives us the probability. Since the table is for 2 DP, we'll look for 0.64 (round to nearest hundredth):

    0.7389

    That's the answer that you're looking for.

    ------

    #2 is exactly the same using different numbers, except you are asked for the proportion that is greater than a given value. So if the table linked above gives you the proportion below a point, and the sum of all proportions under the curve is 1, then you can subtract the "less than" value from 1 to get the remaining "greater than" value.

    Other than that last step, the steps are the same as the first one.

    Hope this helped. Best answer if it does.

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