Math 106 Exam #2
Fall 2002 - Hartlaub
November 8, 2002
To receive full credit you must show your work. The point values associated with each part are clearly marked. Don't spend too much time on one particular problem.
1. If the temperature in Florida falls below 32oF during certain periods of the year, there is a chance that the citrus crop will be damaged. Suppose that the probability is .1 that any given tree will show measurable damage when the temperature falls to 30oF. After the temperature drops to 30oF on a cold January day, 18 trees are randomly selected for inspection.
a. What is the probability that at least 3 trees show measurable damage? (5)
b. What is the probability that at most 3 trees show measurable damage? (5)
c. What is the probability that the first tree showing damage is found while inspecting the fourth tree? (5)
d. What is the expected number of trees showing damage in an orchard of 2000 trees? (5)
e. What is the standard deviation of the number of trees showing damage in an orchard of 2000 trees? (5)
f. What is the probability that more than 10% of the trees in the orchard of 2000 trees will show damage? (5)
2. Let X be the amount of time (in minutes) that a particular San Francisco commuter must wait for a BART train. Suppose that X is uniformly distributed over the interval from 0 to 20.
a. Sketch the density curve for X. (5)
b. What is the probability that X is less than 8 minutes? (5)
c. What is the probability that X is more than 15 minutes? (5)
d. Find the value of c for which P(X < c) = .9. (5)
3. An agronomist examines the cellulose content of a variety of alfalfa hay. Suppose that the cellulose content in the population has standard deviation σ = 8 milligrams per gram (mg/g). A sample of 15 cuttings has mean cellulose content 145 mg/g.
a. Give a 90% confidence interval for the mean cellulose content in the population. (5)
b. How many cuttings would the agronomist need to examine in order to cut the margin of error from part (a) in half? (5)
c. A previous study claimed that the mean cellulose content was μ = 140 mg/g, but the agronomist believes that the mean is higher than that figure. State H0 and Ha and carry out a significance test to see if the new data support this belief. (10)
d. The statistical procedures used in (a) and (c) are valid when several assumptions are met. What are these assumptions? (5)
4. The p-value for a two-sided test of the null hypothesis H0: μ=45 is 0.04.
a. Does the 98% confidence interval include the value 45? Why? (5)
b. Does the 94% confidence interval include the value 45? Why? (5)
5. The amount of money spent by a customer at a discount store has a mean of $35 and a standard deviation of $16. What is the probability that a randomly selected group of 26 shoppers will spend a total of at most $900? (Hint: The total will be at most $900 when the sample average exceeds what value?) (15)
6. An appliance dealer sells three different models of upright freezers having 13.5, 15.9, and 19.1 cubic feet of storage space, respectively. Let X be the amount of storage space purchased by the next customer to buy a freezer. Suppose that X has the following probability distrubution:
x / 13.5 / 15.9 / 19.1P(X=x) / .2 / .5 / .3
a. Calculate the mean and variance of X. (6)
b. If the price of the freezer depends on the size of the storage space, X, in the following way,
Price = 25X-8.5
what is the mean value of the variable price paid by the next customer? (3)
c. What is the variance of the price paid? (3)