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?
To solve these probability questions, we need to use the information given and apply the concepts of probability. Let's start by answering each question step by step.
(a) What is the expected number of parents who contribute (for a given class)?
To find the expected number of parents who contribute in a given class, we need to multiply the percentage of parents contributing to an RESP by the total number of parents.
Given: In 2000, 7.2% of taxpayers with children under the age of 19 contributed to an RESP.
Since there are 20 students in each class, the total number of parents is 20.
Expected number of parents contributing = 7.2% of 20 = (7.2/100) * 20
Therefore, the expected number of parents who contribute for 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 the probability that more than half of parents are contributing in a given class, we need to use the concept of binomial probability.
Given: The government matches 20% of the parent's contributions to an RESP (to an annual maximum).
The probability of a parent contributing to an RESP is 7.2% = 0.072 (from part a).
Using binomial probability formula:
P(X > 10) = 1 - P(X <= 10)
Where:
P(X) represents the probability of X parents contributing.
n represents the total number of parents (20 in this case).
p represents the probability of a parent contributing (0.072 in this case).
Using a binomial probability calculator or software, you can find P(X <= 10) to get the probability of 10 or fewer parents contributing.
For example, using a calculator, putting in the values n = 20, p = 0.072, X = 10, you would find that P(X <= 10) = 0.832 (approximately).
Therefore, P(X > 10) = 1 - P(X <= 10) = 1 - 0.832 = 0.168 (approximately).
Thus, the probability that more than half of parents are contributing in a given class is approximately 0.168.
(c) What is the probability that 1/4 of the classes report more than 3 parents contributing?
To find the probability that 1/4 of the classes report more than 3 parents contributing, we need to use the concept of binomial probability again.
Since there are 16 classrooms, 1/4 of them would be 16 * (1/4) = 4 classrooms.
Using the same formula as before, with n = 20 (total number of parents in a class) and p = 0.072 (probability of a parent contributing), we can calculate P(X > 3) for each classroom and then find the probability for 4 classrooms.
For example, using a calculator, putting in the values n = 20, p = 0.072, X = 3, you would find P(X <= 3) = 0.918 (approximately).
So, P(X > 3) = 1 - P(X <= 3) = 1 - 0.918 = 0.082 (approximately).
The probability of more than 3 parents contributing in each of the 4 classrooms is (0.082)^4 = 0.00003526 (approximately).
Therefore, the probability that 1/4 of the classes report more than 3 parents contributing is approximately 0.00003526.
Please note that the values used in the calculations are approximations for demonstration purposes, and the actual calculations may involve more decimal places for more accurate results.