1. In each of the following, identify the individuals and the variable of interest, and classify the variable as continuous or discrete:
Lengths of boa constrictors:
Boa constrictors/length/continuous
Species of plants living in the southwestern United States:
Plants in southwester United States/species/discrete
Ounces of cereal in boxes:
Cereal boxes/mass/continuous
Price of a gallon of regular gasoline:
Regular gasoline/price/discrete (could be considered as continuous, but the price is never $2.323789532 for one gallon so we say it’s discrete)
Marital status of professional soccer players:
Professional soccer players/marital status/discrete
2. Suppose you know that 60% of registered voters support the current policy of the United States in Afghanistan. If that is the case, and X is the number of registered voters you will have to interview before finding a supporter of this policy, the probability function for X is given by:
x 1 2 3 4 5 6 or more
p(x) .60 .24 .10 .04 .01 .01
a. What is the probability that you find a supporter of the Afghanistan policy in three or fewer tries?
The distribution is geometric so we can either use the table or use the formulas for geometric distribution. Let’s use both!
or
(using the table)
b. How likely is it that at least the first five people you interview do not like the current Afghanistan policy?
or
(using the table)
c. What is the mean (expected value) number of voters you would have to interview to find a supporter of the policy? What is the variance?
The theoretical mean and variance are:
Using the table values and definitions of mean and variance we get:
3. In June 2001, a referendum was held in North Dakota. The question was whether to reverse a previous legislative action which allowed banking companies to employ an opt-out policy regarding their right to sell or otherwise make available information about customers to other companies. An opt-out policy basically forces the customers to actively ask not to have their information distributed by the bank. An opt-in policy forces the banks to ask for permission and not to distribute information unless the customer explicitly says it is okay. In the vote, the legislative action was overturned on a vote of 70% in favor of going back to the previous opt-in law. This action was, to say the least, disappointing to the banks. Suppose the views of the voters of North Dakota are in fact fairly universal, and that a SRS of 2000 people are interviewed.
a. What is the expected number of respondents who will agree with the banks? What is the variance?
b. What is the probability that at least 1350 of the respondents will side with the majority of the voters of North Dakota?
4. In order to gain acceptance to a medical school, a prospective medical student must take (and do well on) a standardized test called the MCAT. The test has four parts, three of which (physical sciences, verbal reasoning, biological sciences) are multiple choice tests whose scaled scores have range 1 to 15 and mean 8. The standard deviation for each of these three sections is 2. The random variables assigning scaled scores on the three parts are all normal distributions. (The fourth part is a writing sample and is not graded on the same sort of scale. We do not consider this part in this question.)
a. Assuming the three random variables P (assigning scaled score on the physical sciences part), V (assigning verbal reasoning scores), and B (assigning biological sciences scores) are independent, what are the mean and standard deviation of the random variable T = P + V + B assigning total score on the three parts?
b. Generally, a total score of at least 30 is needed (along with a high gpa) for admission. What proportion of the students taking the exam would be expected to score 30 or higher?
c. If Tbar is the distribution of all sample means from samples of size 25 drawn from T, what are the mean and standard deviation of Tbar?
d. If the about to graduate group of 25 aspiring medical students at Big State U average a total of 26, would that suggest this school has a stronger pre-med program than average? Explain.
Yes. The observation is 1.67 standard deviations from the mean, which gives a p-value of 0.0475. So we can be confident that this was not due chance.