Question

Suppose a certain capsule is manufactured so that the dosage of the active ingredient follows the distribution Y ~ N(μ = 10 mg, σ = 1 mg). A dosage of 13mg is considered dangerous. Compute the probability that Y falls into the dangerous region.

a. 0.0013.
b. 0.013.
c. 0.9987.
d. 0.
Again suppose a certain capsule is manufactured so that the dosage of the active ingredient follows the distribution Y ∼ N(μ=10, σ=1), where units are in mg. A dosage of 13mg is considered dangerous. If we sample 49 capsules at random, compute the probability that the mean Y-bar falls into the dangerous region and pick the closest answer below.a. 0.0013.
b. 0.013.
c. 0.9987.
d. 0.

Answer 1

(a) Probability that Y falls into the dangerous region is 0.0013.

(b) Probability that the mean Y-bar falls into the dangerous region is 0.00001.

Explanation:

We are given that a certain capsule is manufactured so that the dosage of the active ingredient follows the distribution Y ~ N(μ = 10 mg, σ = 1 mg).

A dosage of 13 mg is considered dangerous.

Let Y = dosage of the active ingredient

The z-score probability distribution for normal distribution is given by;

Z =
`( Y-\mu)/(\sigma) } }` ~ N(0,1)

where,
`\mu` = population mean = 10 mg

`\sigma` = standard deviation = 1 mg

(a) Probability that Y falls into the dangerous region is given by = P(Y
`\geq` 13 mg)

P(Y
`\geq` 13 mg) = P(
`( Y-\mu)/(\sigma) } }`
`\geq`
`( 13-10)/(1) } }` ) = P(Z
`\geq` 3) = 1 - P(Z < 3)

= 1 - 0.9987 = 0.0013

The above probability is calculated by looking at the value of x = 3 in the z table which has an area of 0.9987.

(b) We are given that a dosage of 13 mg is considered dangerous. And we sample 49 capsules at random.

Let
`\bar Y` = sample mean dosage

The z-score probability distribution for sample mean is given by;

Z =
`(\bar Y-\mu)/((\sigma)/(√(n) ) ) } }` ~ N(0,1)

where,
`\mu` = population mean = 10 mg

`\sigma` = standard deviation = 1 mg

n = sample of capsules = 49

So, Probability that the mean Y-bar falls into the dangerous region is given by = P(
`\bar Y`
`\geq` 13 mg)

P(Y
`\geq` 13 mg) = P(
`(\bar Y-\mu)/((\sigma)/(√(n) ) ) } }`
`\geq`
`(13-10)/((1)/(√(49) ) ) } }` ) = P(Z
`\geq` 21) = 1 - P(Z < 21)

= 0.00001