Statistics and Probability Answers

1. The average number of milligrams (mg) of cholesterol in a cup of a certain brand of ice cream is 660 mg, the standard deviation is 35 mg. Assume the variable is normally distributed.

a. If a cup of ice cream is selected, what is the probability that the cholesterol content will be more than 670 mg?

b. If a sample of 10 cups of ice cream is selected, what is the probability that the mean of the sample will be larger than 670 mg?

2.In a study of the life expectancy of 400 people in a certain geographic region, the mean age at death was 70 years, and the standard deviation was 5.1 years. If a sample of 50 people from this region is selected, what is the probability that the mean life expectancy will be less than 68 years?

Average college cost of tuition fee for all private institutions last year was P 36,400. A random sample of costs this year for 45 institutions of higher learning indicated that the sample mean was P 37,900 and sample standard deviation was P 5,600. At the 0.10 level of significance, is there sufficient evidence t0 conclude that the cost has increased?

A report in LTO stated that the average age of taxis in the Philippines is 9 years. An operations manager of a large taxi company selects a sample of 40 taxis and finds the average age of the taxis is 8.2 years. The o if the population is 2.3 years. At a = 0.05, can it be concluded that the average age of the taxis in his company is less than the national average?

A box contains (𝑥+1) red marbles, (2𝑥+4) yellow marbles, and (4𝑥−6) green marbles

(i)

Find an expression, in terms of 𝑥, for the total number of marbles in the box.

(ii)

A marble is drawn at random from the box. Write down an expression, in terms of 𝑥, for the probability that the marble is red.

(iii)

Given that probability in part (ii) is 1/5, find the value of 𝑥.

Suppose three coins are tossed and we are interested to determine the

number of heads that will come out. Let us use H to represent the number

of heads that will come out. List the sample space and count the number of

heads in each outcome and assign this number to this outcome.

2. A company holds a raffle game. The amount to be won are as follows: one Php10, 000; two Php5, 000 each and four Php2, 000 each. Two thousand tickets were sold at Php100 each. If you buy one of the raffle tickets, what is your expected net gain?

 Let X1, X2 have the joint probability density function f(x1, x2) = 2e −(x1+x2) , 0 < x1, x2 < ∞ Let Y1 = X1, Y2 = X2 − X1.

(i) Using the change of variable technique, find the joint probability density function of Y1, Y2

(ii) Find the conditional distribution of Y2 given Y1

The joint probability density function of two random variables X1 and X2 is defined by f(x1, x2, x3) = 2, 0 < x1 < x2 < 1

Find the conditional distribution of X1 given X2 = x

The moment generating function of two jointly distributed random variables X1 and X2 is M(t1, t2) = e ^− 0.5 G where G = (7.51t 2 1 + 7.9t 2 2 + 3.8574t1t2 + 135.4t1 + 137.2t2) Using this function, find the correlation coefficient of of X1 and X2

Let X and Y have joint probability distribution function f(x, y) = ( 2x+y /12) , (x, y) = (0, 1); (0, 2); (1, 2); (1, 3) 0, elsewhere

Find

(i) the covariance between X and Y.

(ii) the joint probability generating function of X and Y.