Suppose X is a normal distribution with N(210, 32). Find the following: a. P( X < 230) b. P(180 < X < 245) c. P( X >190) d. Find c such that P( X < c) = 0.0344 e. Find c such that P( X > c) = 0.7486

(a) The value of P (X < 230) is 0.9998.

(b) The value of P (180 < X < 245) is 1.

(c) The value of P (X > 190) is 0.9998.

(d) The value of c is 199.7.

(e) The value of c is 213.79.

Step-by-step explanation:

It is provided that random variable X follows a Normal distribution with parameters μ = 210 and σ = √32.

(a)

Compute the probability of the event X < 230 as follows:

*Use the z-table for the probability.

Thus, the value of P (X < 230) is 0.9998.

(b)

Compute the probability of the event 180 < X < 245 as follows:

*Use the z-table for the probability.

Thus, the value of P (180 < X < 245) is 1.

(c)

Compute the probability of the event X > 190 as follows:

*Use the z-table for the probability.

Thus, the value of P (X > 190) is 0.9998.

(d)

It is provided that P (X < c) = 0.0344.

The value of z for which P (Z < z) = 0.0344 is -1.82.

Compute the value of c as follows:

Thus, the value of c is 199.7.

(e)

It is provided that P (X > c) = 0.7486.

The value of z for which P (Z < z) = 0.2514 is 0.67.

Compute the value of c as follows:

Thus, the value of c is 213.79.

$6.42

step-by-step explanation:

the revenue of the fitness center, y, is the product of two factors: the number of members and the fee paid by each member. there are 320 members now, but number of members goes down by 10 for each $5 increase in fees, so if x is the number of $5 price increases then members is equal to (320 - 10x). since the fee is currently $45, and the fee increases by $5 for every price bump x, the fee is given by (45 + 5x). so, revenue y is given by y = (320 - 10x)(45 + 5x).