A marketing research firm suggests to a company that two possible competing products can generate incomes X and Y (in millions) that are N(3, 1) and N(3.5, 4), respectively. Clearly, P(X < Y) > 1/2. However, the company would prefer the one with the smaller variance if, in fact, P(X > 2) > P(Y > 2). Which product does the company select?

Find dy/dx for the given equation. $x^{2}-y^{2}=x$

Let X equal the widest diameter (in millimeters) of the fetal head measured between the 16th and 25th weeks of pregnancy. Assume that the distribution of X is N(46.58, 40.96). Let X̅ be the sample mean of a random sample of n = 16 observations of X. a. Give the values of E(X̅) and Var(X̅). b. Find P(44.42 ≤ X̅ ≤ 48.98).

Let X be N(50, 36). Using the same set of axes, sketch the graphs of the probability density functions of a. X. b. X̅, the mean of a random sample of size 9 from this distribution. c. X̅, the mean of a random sample of size 36 from this distribution.

14. Let X1 , X2 , ..., X16 be a random sample from a normal distribution N (77, 25). Compute (a) P(77 < X < 79.5). (b) P(74.2 < X < 78.4) .

What was the connection of the Holocaust to the birth of Israel?

Let Xi be the price (in dollars) of stock i one year from now. X1 is N(15, 100) and X2 is N(20, 2025). Today I buy three shares of stock 1 for $12/share and two shares of stock 2 for$17/share. Assume that X1 and X2 are independent random variables. a. Find the mean and variance of the value of my stocks one year from now. b. What is the probability that one year from now I will have earned at least a 30% return on my investment? c. If X1 and X2 were not independent, why would it be difficult to answer parts (a) and (b)?

A consumer buys n light bulbs, each of which has a lifetime that has a mean of 800 hours, a standard deviation of 100 hours, and a normal distribution. A light bulb is replaced by another as soon as it burns out. Assuming independence of the lifetimes, find the smallest n so that the succession of light bulbs produces light for at least 10,000 hours with a probability of 0.90.

Identify each equation without completing the square.

$$4 x ^ { 2 } + 4 y ^ { 2 } + 12 x + 4 y + 1 = 0$$

Let X equal the weight of the soap in a “6-pound” box. Assume that the distribution of X is N(6.05, 0.0004). a. Find P(X < 6.0171). b. If nine boxes of soap are selected at random from the production line, find the probability that at most two boxes weigh less than 6.0171 pounds each. Hint: Let Y equal the number of boxes that weigh less than 6.0171 pounds. c. Let X̅ be the sample mean of the nine boxes. Find P(X̅ ≤ 6.035).

f(x) = x^2⁄3, [1, 8]

Suppose that the distribution of the weight of a prepackaged “1-pound bag” of carrots is N(1.18, 0.07²) and the distribution of the weight of a prepackaged “3-pound bag” of carrots is N(3.22, 0.09²). Selecting bags at random, find the probability that the sum of three 1-pound bags exceeds the weight of one 3-pound bag. Hint: First determine the distribution of Y, the sum of the three, and then compute P(Y > W), where W is the weight of the 3-pound bag.

14. Let X1 , X2 , ..., X16 be a random sample from a normal distribution N (77, 25). Compute (a) P(77 < X < 79.5). (b) P(74.2 < X < 78.4) .

Let X denote the wing length in millimeters of a male gallinule and Y the wing length in millimeters of a female gallinule. Assume that X is N(184.09, 39.37) and Y is N(171.93, 50.88) and that X and Y are independent If a male and a female gallinule are captured, what is the probability that X is greater than Y?