Question 1144014: Sixty five percent of 12th graders attend school in a particular urban district. If a sample of 450 12th grade children is selected. Find the probability that more than 280 are actually enrolled in the school
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! p = .65
q = 1 - .65 = .35
n = 450
mean = n * p = 450 * .65 = 292.5
s = sqrt (n * p * q) = sqrt ( 450 * .65 * .35) = sqrt (102.375) = 10.118 rounded to 3 decimal places.
the probability that a randomly chosen 12th grader comes from a school in the particular urban district is .65 and the probability that the randomly chosen 12th grade does not come from a school in the particular urban district is 1 - .65 = .35.
you have a mean of 292.5 and a standard error of 10.118.
the standard error is the standard deviation of the distribution of sample means in a theoretically infinite number of samples of size 450.
you want to find the z-score for 280 students attending a school in the particular district.
the z-score formula is z = (x-m)/s.
z is the z-core
m is the mean
s is the standard error.
since m = 292.5 and s = 10.118, and x = 280, the formula becomes:
z = (280-292.5)/10.118.
solve for z to get z = -1.24 rounded to 2 decimal places.
look up a z-score of -1.24 in the normal distribution tables and you will see that the area to the left of that z-score will be .1093.
this means that the area to the right of that z-score will be 1 - .1093 = .8907.
that's your solution.
the probability that more than 280 are actually enrolled in the school is .8907.
i used rounded numbers and i used the normal distribution table rather than a normal distribution calculator.
if i had used the normal distribution calculator in my TI-84 Plus amd i didn't round the number, i would have gotten .891661987.
either one of these answers should be sufficient depending on what rounding your instructor expects from you for intermediate results and for final results.
i also did the same problem using an online calculator that has its own rounding rules.
the answer i got using that calculator was .8917.
take your pick.
if you were given tools to do the calculations for you, then use those tools.
the procedure, however, should be the same.
n = 450
p = .65
q = .35
m = 292.5
s = sqrt(450 * .65 * .35) = 10.11805317 with calculator display rounding only.
z = (280 - 292.5) / 10.11805317 = -1.235415528 with calculator display rounding only.
you would then find the area to the left of that z-score.
you would then find the area to the right of that z-score.
the area to the right of that z-score is the probability that more than 280 students are actually enrolled in that school.
this assumes that i did this correctly, of course.
i'm pretty sure i did, if my memory serves me correctly.
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