Question

The amount of time that a 4th grader reads comic books in a week is normally distributed with a mean of 6 hours and a standard deviation of 2 hours. Suppose a 4th grader is considered addicted if he/she reads comic books for more than 11 hours a week. If twenty 4th graders are to be chosen at random, find the probability that at least one of them is addicted to comic books (round off to second decimal place).

Answer 1

Answer:

The probability that at least one of the 20 4th graders is addicted to comic books is 0.12.

Step-by-step explanation:

Let X = number of hours a 4th grader reads comics and Y = number of 4th grader addicted to reading comics.

The random variable X is continuous and follows a normal distribution with mean, μ = 6 hours and standard deviation, σ = 2 hours.

And the random variable Y is discrete and follows a binomial distribution with success defined as a 4th grader is addicted to reading comics.

Compute the probability that a 4th grader is addicted to reading comics as follows, i.e. determine the value of P (X > 11) .

$P(X>11) = P(\frac{X-\mu}{\sigma} >\frac{11-6}{2})\\=P(Z >2.5)\\=1-P(Z<2.5)\\=1-0.9938\\=0.0062$

Use the Z-table for the probability value.

Now compute the probability that out of 20 fourth graders at least 1 is addicted to comics as follows:

The Binomial probability function is:

$P(Y)=^{n}C_{y} p^{y} (1-p)^{n-y}$

Compute the value of $P(Y\geq 1)$ as follows:

$P(Y\geq 1)=1-P(Y<1)\\=1-P(Y=0)\\=1-[^{20}C_{20} (0.0062)^{0} (1-0.0062)^{20-0}\\=1-(1\times1\times0.88304)\\=0.11696\\\approx0.12$

Thus, the probability that at least one of the 20 4th graders is addicted to comic books is 0.12.

Answer 2

The probability that at least one of them is addicted to comic books is; P(X ≥ 1) = 0.12

We are told that this question is a normal distribution with;

Mean; μ = 6 hours

Standard deviation; σ = 2 hours

The formula for the z-score of a normal distribution is;

z = (x' - μ)/σ

Now, we are given that the 4th grader is considered addicted if he/she reads comic books for more than 11 hours a week. Thus;

x' = 11 hours

Thus;L, probability that a 4th grader is addicted to comic books is; P(X > 11)

z = (11 - 6)/2

z = 2.5

From online p-value from z-score calculator, we have; P(X > 11) = 0.00621

To find the probability that at least one of them is addicted to comic book, we will use binomial probability distribution with the formula;

P(X = x) = ⁿCx × p^(x) × q^(n - x)

Where q = 1 - p = 1 - 0.00621

q = 0.99379

Thus, the probability that at least one of them is addicted to comic book if 20 first graders are chosen at random is;

P(X ≥ 1) = 1 - P(X = 0)

P(X = 0) = 20C20 × 0.00621^(0) × 0.99379^(20 - 0)

P(X = 0) = 0.8829

Thus;

P(X ≥ 1) = 1 - 0.8829

P(X ≥ 1) = 0.1171

Approximating to 2 decimal places gives;

P(X ≥ 1) = 0.12

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