of all 3-to 5-year- old children, 56% are enrolled in school. If a sample of 500 such children is randomly selected, find the probability that at least 250 will be enrolled in school.
Find mean and standard deviation. Use z-scores to find z, then a z-table to find your probability based on the z-score. This is a normal approximation to the binomial distribution.
I'll get you started:
mean = np = (500)(.56) = ?
standard deviation = √npq = √(500)(.56)(.44) = ?
Note: q = 1 - p
Finish the calculations.
To find z-score:
z = (x - mean)/sd
Note: x = 250
Once you have the z-score, look in a z-table using the score to find your probability. Remember that the question is asking "at least 250" when looking at the table.
To find the probability that at least 250 children will be enrolled in school, we need to calculate the probability of having exactly 250, 251, 252, ..., 500 children enrolled and then add up these probabilities.
Let's start by finding the probability of having exactly 250 children enrolled in school.
P(250 children enrolled) = (0.56)^250 * (1-0.56)^(500-250) * (500 choose 250)
where (0.56)^250 represents the probability that a child is enrolled, (1-0.56)^(500-250) represents the probability that a child is not enrolled, and (500 choose 250) represents the number of ways to choose 250 children out of 500.
We can use a similar calculation for the probabilities of having 251, 252, ..., 500 children enrolled.
Finally, we sum up all these probabilities to get the probability that at least 250 children will be enrolled in school.
P(at least 250 children enrolled) = P(250 children enrolled) + P(251 children enrolled) + ... + P(500 children enrolled)
To find the probability that at least 250 children out of 500 are enrolled in school, we can use the binomial probability formula.
The binomial probability formula is given by:
P(x) = (n C x) * (p^x) * ((1-p)^(n-x))
Where P(x) is the probability of x success events, n is the total number of trials, p is the probability of a success event, (n C x) represents the combination of n items taken x at a time, and ^( ) represents exponentiation.
In this case, the total number of trials is 500, the probability of a child being enrolled in school is 0.56, and we want to find the probability of at least 250 children being enrolled. This means we need to calculate the probability of 250, 251, 252, 253, ..., 500 children being enrolled and then sum them up.
P(at least 250 children enrolled) = P(250) + P(251) + P(252) + ... + P(500)
To calculate each individual probability, we substitute the respective values into the formula.
P(250) = (500 C 250) * (0.56^250) * ((1-0.56)^(500-250))
P(251) = (500 C 251) * (0.56^251) * ((1-0.56)^(500-251))
...
P(500) = (500 C 500) * (0.56^500) * ((1-0.56)^(500-500))
Since calculating each individual probability might be time-consuming, we can use a statistical software or spreadsheet to calculate the probabilities automatically.