Download Math 54 – Final Review

Math 54 – Final Review
Topic: Probability distributions.
1. Determine whether each of the following tables represents a probability distribution.
If the table is not a probability distribution, explain why.
a.
___x____|_____p(x)___
0
0.4219
1
0.4219
2
0.1406
3
0.0156
b.
___x____|_____p(x)___
0
0.502
1
0..365
2
0.098
3
0.011
4
0.001
2. For the table that is a probability distribution part 1, answer the following questions.
a. What is the probability of x = 1 or x = 2.
b. Is the value of x = 3 unusual? Why or why not?
c. Find the mean and the standard deviation of the probability distribution.
d. Interpret the mean.
3. An insurance company charges a 21-year-old male $250 for a one year $100,000 life
insurance policy. A 21-year-old male has a probability of 0.9985 of living one year.
a. What is the probability that a 21-year-old male dies in the next year?
b. If the purchaser dies in the next year, what is the financial result of this purchase for
his estate?
c. If the purchaser lives through the year, what is the financial result of this purchase for
him?
d. Set up a probability distribution representing this situation.
e. Find the expected value of this life insurance policy and interpret the result.
Topic: The binomial distribution
1. What is the 4-part definition of a binomial distribution?
2. What is the binomial probability formula?
3. What are the two calculator functions for calculation binomial probabilities and how are they each used?
4. The CBS television show 60 Minutes recently had a market share of 20, meaning that 20% of all
households watching TV were tuned to 60 Minutes. A random sample of 10 households with TV sets in
use at the time of 60 Minutes is selected.
a. Why is this a binomial distribution?
b. Using the binomial probability formula, find the probability that exactly 5 of the households were
watching 60 Minutes.
Solve the remaining parts using the appropriate calculator function.
c. What is the probability that less than 5 of the households were watching 60 minutes?
d. What is the probability that 6 or more of the household were watching 60 Minutes?
e. Would it be unusual for 6 or more households to be watching 60 Minutes? Why?
5. In a clinical trial of Lipitor, a common drug, used to lower cholesterol, 863 patients were given a
treatment of Lipitor tablets. That group consisted of 19 patients who experiences flu symptoms.
a. Given that the probability of that a randomly chosen person is experiencing flu symptoms is 0.019, find
the mean and standard deviation of the number of people in groups of 863 that can be expected to have flu
symptoms. (This refers to the general population, not just those taking the drug.)
b. Based on the result from part a, find the interval that would represent the usual range for the number of
people in groups of 863 who would be experiencing flu symptoms.
c. Using your interval from b, is it unusual for 19 patients in a group of 863 to experience flu symptoms?
Why or why not.
d. Based on your result from c, do flu symptoms appear to be an adverse reaction that should be of concern
to those that take Lipitor?
Topic: Normal Distributions
1. What is the definition of the standard normal distribution?
2. Using table A-1 find each of the following probabilities.
a. P( z > 1.45)
b. P(-2.34 < z < 1.06)
c. P(z < -0.56)
3. Find z for each the following conditions.
a. z that corresponds to P35, the 35th percentile.
b. z that separates the top 10% of values from the rest.
c. z such that the area between –z and z is 0.8664
4. Birth weights in the United States are normally distributed with a mean of 3420 grams and a standard
deviation of 495 grams.
a. What is the probability that a baby is born weighing less than 3000 grams? Use the formulas and tables
to solve. Show all work.
Use the appropriate calculator function to solve the following.
b. What is the probability that a baby is born weighing more than 4500 grams.
c. What percentage of babies are born weighing between 2500 and 4500 grams?
d. If a hospital wants to set up special procedures for the smallest 2% of babies, what weight is used for the
cut-off separating the lightest 2% from the rest?
Topic: Sampling distributions
1. What is the definition of the sampling distribution of the proportion?
2. Under what conditions can this distribution be considered normal?
3. What is the definition of the sampling distribution of the mean?
4. Under what conditions can this distribution be considered normal?
5. An exam consists of 50 multiple choice questions. Based on how much you have studied, you think you
have a probability of 0.70 of getting the correct answer. Consider the sampling distribution of the sample
proportion of the 50 questions on which you get the correct answer.
a. Find and interpret the mean of this sampling distribution.
b. Find and interpret the standard error of this sampling distribution.
c. What do you expect for the shape of the sampling distribution? Why?
d. If truly p=0.70, would it be unusual for you to answer 60% of the questions correctly?
6. In an exit poll of 2705 People, 1528 voted yes for a certain proposition. Is this enough to decide that
proposition passed?
a. What is the point estimate of the proportion of people who voted yes for this proposition based on this
sample. Is this a p or a p-hat value?
b. Assume that the proportion who voted yes was p = 0.50. Find standard error based on this assumption.
c. Find the z-score for the point estimate found in a. Does this z-score provide strong evidence that the
proposition will pass?
7. The scores on the Psychomotor Development Index (PDI), a scale of infant development, have a normal
distribution with a mean of 100 and a standard deviation of 15.
a. One infant is selected and has a PDI value of 90, find the z-score for this infant.
b. A study uses a random group of 225 infants. Find the z-score for a mean PDI score of 90 for this group
of infants.
c. Explain why a PDI value of 90 for a single infant is not surprising, but a mean PDI score of 90 for a
group of 225 infants would be surprising.