Question 1191669: College students often make up a substantial portion of the population of college cities and towns. State College, Pennsylvania, ranks first with 71.1% of its population made up of college students. What is the probability that in a random sample of 171 people from State College, more than 50 are not college students (rounded to 4 decimal places)?
P(X>50) =
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! proportion of people who attend college in the population is assumed to be equal to .711.
if that is true, then the proportion of people who do not attend college in the population is assumed to be equal to 1 minus that = .289.
in a sample of 171 students from this population, the question that is being asked is:
what is the probability that more than 50 of the people in that sample do not attend college?
the sample proportion becomes 50 / 171 = .292398 rounded to 6 decimal places.
the standard error that i believe is recommended to be used for this calculation is equal to sqrt(p * q / n).
the p and the q are taken from the population proportions.
that makes the standard error equal to sqrt(.711 * .289 / 171) = .034665 rounded to 6 decimal places.
note that q is equal to (1 - p).
the z-score formula is used.
that formula says that z = (x - m) / s
z is the z-score
x is the sample proportion that is to be compared to the population proportion.
m is the population proportion.
s is the standard error.
you get z = (x - m) / s becomes z = (.292398 - .289) / .034665 = .098024 rounded to 6 decimal places.
a z-score of .098024 has an area under the normal distribution curve equal to .460957 to the right of it when rounded to 6 decimal places.
round that to 4 decimal places to get an area under the normal distribution curve equal to .4610 to the right of it.
that means that the probability that more than 50 people in a sample of 171 do not attend college is .4610.
try this out.
see if it works.
let me know if it does or doesn't.
theo
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