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.

(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.

Step-by-step 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  =    ~ N(0,1)

where, = population mean = 10 mg

            = standard deviation = 1 mg

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

       P(Y 13 mg) = P( ) = P(Z 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 = sample mean dosage

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

                                 Z  =    ~ N(0,1)

where, = population mean = 10 mg

            = 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(13 mg)

          P(Y 13 mg) = P( ) = P(Z 21) = 1 - P(Z < 21)  

                                                             = 0.00001

the entire expression should be subtracted from 30. to subtract the quantity, distribute the negative sign, which changes the signs on the terms. so, the –2 changes to positive and gets added to 30 instead of subtracted from it.

hope this ! ^w^

1) is right

step-by-step explanation: