Part 1-
Consider the following scenario:
Let P(C) = 0.4.
Let P(D) = 0.5.
Let P(C|D) = 0.6.
a. Find P(C AND D).
b. Are C and D mutually exclusive? Why or why not?
c. Are C and D independent events? Why or why not?
d. Find P(C OR D).
e. Find P(D|C).
Part 2-
1. A school newspaper reporter decides to randomly survey 12 students to see if they will attend Tet (Vietnamese New Year) festivities this year. Based on past years, she knows that 18% of students attend Tet festivities. We are interested in the number of students who will attend the festivities.
a. In words, define the random variable X.
b. List the values that X may take on.
c. Give the distribution of X.
d. How many of the 12 students do we expect to attend the festivities?
e. Find the probability that at most four students will attend.
f. Find the probability that more than two students will attend.
2. The switchboard in a Minneapolis law office gets an average of 5 incoming phone calls during the noon hour. Experience shows that the existing staff can handle up to six calls in an hour. Let X = the number of calls received during the noon hour.
a. Find the mean and standard deviation of X.
b. What is the probability that the office receives at most six calls at noon on Monday?
c. Find the probability that the law office receives exactly six calls at noon. What does this mean to the law office staff who gets, on average, 5 incoming phone calls at noon?
d. What is the probability that the office receives more than eight calls at noon?
Part 3-
According to a study by Dr. John McDougall of his live-in weight loss program at St. Helena Hospital, the people who follow his program lose between six and 15 pounds a month until they approach trim body weight. Let's suppose that the weight loss is uniformly distributed. We are interested in the weight loss of a randomly selected individual following the program for one month.
a. Define the random variable. X =
b. X ~
c. Graph the probability distribution.
d. f(x) =
e. µ =
f. σ =
g. Find the probability that the individual lost more than ten pounds in a month.
h. Suppose it is known that the individual lost more than ten pounds in a month. Find the probability that he lost less than 12 pounds in the month.
i. P(7 < x < 13|x > 9) = ______ . State this in a probability question, similarly to parts g and h, draw the picture, and find the probability.
Part 4-
1. IQ is normally distributed with a mean of 100 and a standard deviation of 15. Suppose one individual is randomly chosen. Let X = IQ of an individual.
a. Find the probability that the person has an IQ greater than 120. Include a sketch of the graph, and write a probability statement.
b. MENSA is an organization whose members have the top 2% of all IQs. Find the minimum IQ needed to qualify for the MENSA organization. Sketch the graph, and write the probability statement.
c. The middle 50% of IQs fall between what two values? Sketch the graph and write the probability statement.
2. Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 250 feet and a standard deviation of 50 feet.
a. If X = distance in feet for a fly ball, then X ~ _____( ___ ,___ )
b. If one fly ball is randomly chosen from this distribution, what is the probability that this ball traveled fewer than 220 feet? Sketch the graph. Scale the horizontal axis X. Shade the region corresponding to the probability. Find the probability.
c. Find the 80th percentile of the distribution of fly balls. Sketch the graph, and write the probability statement.
Part 5-
Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 250 feet and a standard deviation of 50 feet. We randomly sample 49 fly balls.
a. If X¯ = average distance in feet for 49 fly balls, then X¯ ~ ___ (____ , ____ )
b. What is the probability that the 49 balls traveled an average of less than 240 feet? Sketch the graph. Scale the horizontal axis for X¯. Shade the region corresponding to the probability. Find the probability.
c. Find the 80th percentile of the distribution of the average of 49 fly balls.
Part 6-
1. From generation to generation, the mean age when smokers first start to smoke was believed to be 17. A survey of 40 smokers of the millennial generation was done to see if the mean starting age is now different. The sample mean is 18.1 with a sample standard deviation of 1.3. Do the data support that the starting age is now different at the 5% level? Construct a 95% confidence interval for the true age and interpret it.
2. An article in the San Jose Mercury News stated that students in the California state university system take 4.5 years, on average, to finish their undergraduate degrees. Suppose you believe that the mean time is longer. You conduct a survey of 49 students and obtain a sample mean of 5.1 with a sample standard deviation of 1.2. Do the data support your claim at the 1% level?