Practice Problems for EXAM 2
Be sure to show your work and explain your reasoning (using methods, ideas, and/or terms from our course).
I. Suppose that the probability that Tom Brady completes any one pass is 0.619. Suppose that Tom attempts 9 passes in a quarter of a football game.
a) Calculate the probability that Brady completes at least one of his next 9 passes. Show your work.
b) On what 2 assumptions is your work in part (a) based?
Give a critique of both assumptions in this situation.
II. The Reduced Risk Company has probability 0.92 of processing any single insurance claim correctly. Suppose that Laura has just been hired by RR as an additional claims-processor. For quality control, Laura’s boss will randomly select 150 of the claims that Laura has processed (after training is over) and calculate the proportion of the 150 sampled claims which Laura has processed correctly.
a) IF Laura’s probability of processing a claim correctly is .92, specify the approximate sampling distribution for the proportion of correctly processed claims in the sample of 150. Be sure to include the sampling distribution’s…
i. SHAPE
ii. CENTER (by computing its mean)
iii. SPREAD (by computing its standard deviation)
b) Check each criterion needed to justify your answer to part (a).
c) Assuming that Laura’s probability of processing a claim correctly is .92, calculate the (approximate) probability that she will process less than 86% of the 150 sample claims correctly. Use your answer to part (a) and show your work.
The Reduced Risk Insurance Company has 1000 employees. The following table shows how they are categorized by gender and by company role.
Manager / Sales / Clerical / AccountingMale / 80 / 420 / 10 / 90
Female / 20 / 180 / 140 / 60
III. Suppose that you choose an employee at random from among all employees of the Reduced Risk Company (RR). [See the above information about RR.]
a) Calculate the probability that the employee you choose is female.
b) Justify your calculation in part (a).
c) Calculate the probability that the employee you choose is a female manager.
d) Calculate the probability that the employee you choose is female or a manager. Show your work.
e) If you choose a female employee at random, what is the value of the probability that the chosen female is a manager? Show your work.
f) Are the events “randomly chosen employee is female” and “randomly chosen employee is a manager” mutually exclusive events? Explain why or why not.
g) Are the events “randomly chosen employee is female” and “randomly chosen employee is a manager” independent events? Explain why or why not.
IV.Suppose that Taylor’s former basketball player (and Math major) RJ Beucler tends to make 85% of the free throws he shoots. If he shoots 15 free throws in a practice session, let X= the number of free throws, out of the 15, that RJ makes.
a) What type of distribution does X have? (Specify the family.) Justify that any/all criteria are met for your answer.
b) What does the expected value (a.k.a. mean) equal for X? Show your work.
c) What does the standard deviation equal for X? Show your work.
V. The local branch office of the Reduced Risk Company (RR) has 3 male employees and 2 female employees. Suppose they sample two employees at random (& withOUT replacement) from among the branch office’s employees.
Let X= the number of females in your sample of size 2.
Give a table for the probability distribution function of this X. Show your supporting calculations.
VI. The following is a graph of a probability density function for a continuous random variable X.
a) Calculate the probability that X is less than 1.5. Show your work.
b) What is the value of the mean for X? Justify your answer.
c) What is the value of the median for X? Justify your answer.
VII. Let Y = the number of males in a randomly chosen sample of five employees from the entire RR company. Here is a gift from me to you: the probability distribution function for Y is given (approximately) by the following table:
y / p(y)0 / .01
1 / .08
2 / .23
3 / .34
4 / .26
5 / .08
a) Compute the probability that there are 2 or more males in the sample. Show your work.
b) Compute the expected number E(Y) of males in the sample. [This is also called the mean of Y.] Show your work.
VIII. Here are some facts to use about the population of all runners who finished the 2000 BayState marathon:
- the mean of the population’s finishing times is 228 minutes
- the standard deviation of the population’s finishing times is 35 minutes
The method I used to gather the class dataset was to sample 50 of the finishers at random. Then I calculate the mean of the sample.
a) Specify the approximate sampling distribution for the mean of the 50 sample finishing times. Be sure to include the sampling distribution’s…
i. SHAPE
ii. CENTER (by computing its mean)
iii. SPREAD (by computing its standard deviation)
b) Check each criterion needed to justify your answer to part (a).
c) Calculate (approximately) the 20th percentile for the sampling distribution from part (a). Show your work.
d) For my actual sample of 50 runners, the sample mean was 229.5 minutes and the sample standard deviation was 37.0 minutes. Calculate the sample mean’s standard error for this particular sample. Show your work.