Question 1120938: Please help me solve this probability question: Find the probability that of the next 200 children born, (a) less than 40% will be boys, (b) between 43% and
57% will be girls, and (c) more than 54% will be boys. Assume equal probabilities for the births of boys and
girls.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the probability any given child will be a boy is .5
the probability any given child will be a girl is .5
you sample 200 children from a population.
if you are looking at this from the boys' perspective, the mean of the sample will be p * n = .5 * 200 = 100 boys.
the standard deviation of the sample will be square root of (n * p * q) which is equal to square root of (200 * .5 * .5) = square root of 50 = 7.071067812.
q mentioned in the above formula is the probability that the child will not be a boy.
q is equal to 1 - p which is equal to .5.
so, you have:
mean = 100
standard deviation (actually standard error) = 7.071067812.
since a normal distribution is assumed, you would use the z-score formula to find what you need.
that formula is:
z = (x-m) / s
z is the z-score
x is the raw score
m is the mean
s is the standard error
if you want the probability that less than 40% will be boys, you need to get the z-score that has an area to the left of it of 40% under the nomral distribution curve.
using my TI-86 Plus calculator, i find that the z-score is -.2533471011.
to find the raw score associated with that, i use the z-score formula.
z = (x - m) / s becomes -.2533471011 = (x - 100) / 7.071067812.
solve for x to get x = -.2533471011 * 7.071067812 + 100 = 98.20856547.
from a z-score perspective, it looks like this:
from a raw score perpsective, it looks like this:
any differences from the numbers i've shown using my TI-84 Plus has to do with rounding differences between the TI-84 Plus and the online calculator that gives the display.
the TI-84 Plus keeps more of the decimal digits in the calculation than the online calculator does, so there is some truncation going on.
if we want to find the probability that between 43% and 57% will be girls, we can do the following.
p = .5 = probability that any randomly selected child will be a girl.
q = 1 - .5 = .5 = probability that any randomly selected child will be a boy.
out of 200 children born, 100 are expected to be girls.
that would be n * p = mean which becomes 200 * .5 = 100.
the standard error is equal to square root of (p * q * n) which is equal to square root of (.5 * .5 * 200) which is equal to square root of (50) which is equal to 7.071067812.
we again look for z-scores.
the z-score that has an area under the normal distribution curve of 43% to the left of it would be equal to -.1763741565
the z-score that has an area under the normal distribution curve of 57% to the left of it would be equal to +.1763741565
the area between those twp z-scores will be 57% minus 43% = 14%.
note that the areas i'm showing are percent areas.
the areas use by the z-score calculator are the decimal equivalents.
43% decimal equivalent = .43
57% decimal equivalent = .57
14% decimal equivalent = .14
to find the raw scores, you again use the z-score formula.
that formula is z = (x - m) / s
for the low z-score, you get:
-.1763741565 = (x - 100) / 7.071067812 = 98.75284638
for the high z-score, you get:
+.1763741565 = (x - 100) / 7.071067812 = 101.2471536
there is a 14% probability that the number of girls will be between 98.75284638 and 101.2471536.
this can be shown visually below.
using z-scores:
using raw scores:
when you are looking for the probability that more than 54% will be boys, you look for the probability that less than 100% - 54% = 46% will be boys.
the same z-score will apply, because area to the left of 46% means 100% - 46% = 54% to the right.
i'll show this using the online calculator.
using the TI-84 Plus, i get 46% are to the left of the z-score gives me a z-score of -.1004337118.
use the z-score formula to find the raw score.
z = (x - m) / s becomes -.1004337118 = (x - 100) / 7.071067812.
solve for x to get x = -.1004337118 * 7.071067812 + 100 = 99.28982641.
you can see this visually using the online calculator.
using z-scores, you get:
using raw scores, you get:
those were the area to the left of the z-score and raw score.
areas to the right of the z-score and raw score are shown below.
to the left was 46%.
to the right will be 54%.
using z-scores:
using raw scores:
note that people must be integers.
i left the answers as they were because there was more accuracy that way, but people have to be whole numbers and so you might want to round to the nearest integer under the assumption that the percentages won't be exactly as required, but close.
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