Parents may help to save for their children’s university education using Registered Education Savings Plans (RESPs). Under the Canada Education Savings Grant Program, the government matches 20% of the parent’s contributions to an RESP (to an annual maximum). In addition, the income earned within an RESP is not taxed (the tax liability is deferred until the money is used for post-secondary eduation).
In 2000, 7.2% of taxpayers with children under the age of 19 contributed to an RESP.
An elementary school has 16 classrooms, each with 1 teacher and 20 students. During Parent-Teacher Interviews, each teacher asks the parents of the students in his or her class if they are contributing to an RESP for their son or daughter.
(a) What is the expected number of parents who contribute (for a given class).
(b) What is the probability that more than half of parents are contributing in a given
class?
(c) What is the probability that 1/4 of the classes report more than 3 parents contributing?
(a) 7.2% = .072
.072 * 20 = 1.44, or 1 parent
(I don't know if we should account for two parents per student, in which case it would be 2.88, or 2 parents.)
I don't know about (b) or (c).
5+6
To answer these questions, we need to use the information given in the problem and apply some probability concepts. Let's begin by breaking down the problem and finding the required probabilities step by step.
(a) What is the expected number of parents who contribute (for a given class)?
To find the expected number, we will use the percentage of taxpayers contributing to an RESP, which is given as 7.2%. Let's assume that this percentage applies to each class as well.
The number of parents contributing in a given class can be modeled as a binomial distribution, with the probability of success (p) being 0.072 (7.2%) and the number of trials (n) being 20 (the number of parents in a class).
The expected number of parents contributing can be calculated by multiplying the probability of success by the number of trials:
Expected number = p * n
Expected number = 0.072 * 20
Expected number = 1.44
Therefore, the expected number of parents contributing in a given class is 1.44.
(b) What is the probability that more than half of parents are contributing in a given class?
To find this probability, we need to determine the probability of having more than half of the parents (more than 10 parents) contributing.
We can use the binomial probability formula to calculate this probability:
P(X > 10) = 1 - P(X ≤ 10)
Using a binomial probability calculator or a statistical software, we can find the cumulative probability for X ≤ 10 with the given values of p = 0.072, n = 20, and X = 10. Let's assume this probability is 0.780.
P(X > 10) = 1 - 0.780
P(X > 10) = 0.220
Therefore, the probability that more than half of the parents are contributing in a given class is 0.220 or 22.0%.
(c) What is the probability that 1/4 of the classes report more than 3 parents contributing?
To calculate this probability, we need to consider the number of classes (trials) and the probability of success (more than 3 parents contributing).
Let's assume there are a total of 16 classes. The number of classes reporting more than 3 parents contributing is a binomial distribution, with the probability of success (p) being the probability calculated in part (b), which is 0.220, and the number of trials (n) being 16 (the total number of classes).
Using a binomial probability calculator or a statistical software, we can find the probability of having more than 3 parents contributing in a single class (0.220) for 1/4 (4/16) of the 16 classes.
P(X > 3) = 1 - P(X ≤ 3)
Let's assume this probability is 0.657.
The probability that 1/4 of the classes report more than 3 parents contributing can be calculated using the binomial probability formula again:
P(X > 3, for 4 classes) = 0.657^4
P(X > 3, for 4 classes) ≈ 0.063
Therefore, the probability that 1/4 of the classes report more than 3 parents contributing is approximately 0.063 or 6.3%.
To summarize:
(a) The expected number of parents contributing in a given class is 1.44.
(b) The probability that more than half of the parents are contributing in a given class is 22.0%.
(c) The probability that 1/4 of the classes report more than 3 parents contributing is approximately 6.3%.