Decide whether you can use the normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use the binomial distribution to find the indicated probabilities. Five percent of workers in a city use public transportation to get to work. You randomly select 254 workers and ask them if they use public transportation to get to work. Complete parts (a) through (d).
Can the normal distribution be used to approximate the binomial distribution?
A.
Yes, because both
np greater than or equals≥5
and
nq greater than or equals≥5.
B.
No, because
nq less than<5.
C.
No, because
np less than<5.
(a) Find the probability that exactly
1919
workers will say yes.
What is the indicated probability?
nothing
(Round to four decimal places as needed.)
Sketch the graph of the normal distribution with the indicated probability shaded.
(b) Find the probability that at least
77
workers will say yes.
What is the indicated probability?
nothing
(Round to four decimal places as needed.)
Sketch the graph of the normal distribution with the indicated probability shaded.
A.
1317x
mu equals 12.7μ=12.7
x y graph
B.
1317x
mu equals 12.7μ=12.7
x y graph
C.
1317x
mu equals 12.7μ=12.7
x y graph
D.
The normal distribution cannot be used to approximate the binomial distribution.
(c) Find the probability that fewer than
1919
workers will say yes.
What is the indicated probability?
nothing
(Round to four decimal places as needed.)
Sketch the graph of the normal distribution with the indicated probability shaded.
A.
13119x
mu equals 12.7μ=12.7
x y graph
B.
13119x
mu equals 12.7μ=12.7
x y graph
C.
13119x
mu equals 12.7μ=12.7
x y graph
D.
The normal distribution cannot be used to approximate the binomial distribution.
(d) A transit authority offers discount rates to companies that have at least
3030
employees who use public transportation to get to work. There are
499499
employees in a company. What is the probability that the company will not get the discount?
Can the normal distribution be used to approximate the binomial distribution?
A.
No, because
npless than<5.
B.
Yes, because both
npgreater than or equals≥5
and
nqgreater than or equals≥5.
C.
No, because
nqless than<5.
What is the probability that the company will not get the discount?
nothing
(Round to four decimal places as needed.)
Sketch the graph of the normal distribution with the indicated probability shaded.
In: Math
The January 1986 mission of the Space Shuttle Challenger
was the 25th such shuttle mission. It was unsuccessful due to an
explosion caused by an O-ring seal failure.
(a) According to NASA, the probability of such a failure in a single mission was 1/60,158. Using this value of p and assuming all missions are independent, calculate the probability of no mission failures in 26 attempts. Then calculate the probability of at least one mission failure in 26 attempts. (Do not round your intermediate calculation and round your final answers to 4 decimal places.)
| P(x = 0) | |
| P(x ≥ 1) | |
(b) According to a study conducted for the Air Force, the probability of such a failure in a single mission was 1/35. Recalculate the probability of no mission failures in 26 attempts and the probability of at least one mission failure in 26 attempts. (Do not round your intermediate calculation and round your final answers to 4 decimal places.)
| P(x = 0) | |
| P(x ≥1) | |
(c) Based on your answers to parts a and b, which value of p seems more likely to be true? Explain. (Enter answer as a division like x/y.)
p 1/ 60,158 or 1/35 or
1/36
(d) How small must p be made in order to ensure that the probability of no mission failures in 26 attempts is .999? (Round your answers to 5 decimal places.)
| q | |
| P | |
In: Statistics and Probability
An experiment is given together with an event. Find the (modeled) probability of each event, assuming that the coins are distinguishable and fair, and that what is observed are the faces uppermost.
Three coins are tossed; the result is at most one tail.
An experiment is given together with an event. Find the (modeled) probability of each event, assuming that the dice are distinguishable and fair, and that what is observed are the numbers uppermost.
Two dice are rolled; the numbers add to 3.
An experiment is given together with an event. Find the (modeled) probability of each event, assuming that the dice are distinguishable and fair, and that what is observed are the numbers uppermost.
Two dice are rolled; the numbers add to 11.
An experiment is given together with an event. Find the (modeled) probability of each event, assuming that the dice are distinguishable and fair, and that what is observed are the numbers uppermost.
Two dice are rolled; the numbers add to 13.
An experiment is given together with an event. Find the (modeled) probability of each event, assuming that the dice are distinguishable and fair, and that what is observed are the numbers uppermost.
Two dice are rolled; both numbers are prime. (A positive integer is prime if it is neither 1 nor a product of smaller integers.)
Use the given information to find the indicated probability.
P(A ∪ B) = .8, P(B) = .7, P(A ∩ B) = .4.
Find P(A).
Use the given information to find the indicated probability.
P(A) = .78.
Find P(A').
P(A') =
In: Advanced Math
A department store sells sport shirts in three sizes (small, medium, and large), three patterns (plaid, print, and stripe), and two sleeve lengths (long and short). The accompanying tables give the proportions of shirts sold in the various category combinations.
| Short-sleeved | |||
|---|---|---|---|
| Pattern | |||
| Size | Pl | Pr | St |
| S | 0.04 | 0.02 | 0.05 |
| M | 0.08 | 0.05 | 0.12 |
| L | 0.03 | 0.07 | 0.08 |
| Long-sleeved | |||
|---|---|---|---|
| Pattern | |||
| Size | Pl | Pr | St |
| S | 0.03 | 0.02 | 0.03 |
| M | 0.06 | 0.11 | 0.07 |
| L | 0.04 | 0.02 | 0.08 |
(a) What is the probability that the next shirt sold is a
medium, long-sleeved, print shirt?
(b) What is the probability that the next shirt sold is a medium
print shirt?
(c) What is the probability that the next shirt sold is a
short-sleeved shirt? A long-sleeved shirt?
| short-sleeved shirt | ||
| long-sleeved shirt |
(d) What is the probability that the size of the next shirt sold is
medium?
What is the probability that the pattern of the next shirt sold is
a print?
(e) Given that the shirt just sold was a short-sleeved plaid, what
is the probability that its size was medium? (Round your answer to
three decimal places.)
(f) Given that the shirt just sold was a medium plaid, what is the
probability that it was short-sleeved? Long-sleeved? (Round your
answer to three decimal places.)
| short-sleeved | ||
| long-sleeved |
In: Statistics and Probability
It is suggested that the average IQ of top civil servants,
research scientists and professors is 140. Suppose that the
standard deviation is 5.
a. Suppose a full professor from a Canadian university is selected
at random. What is the probability that the IQ of the selected
Canadian professor is below 130? State any necessary assumptions
you have made to compute this probability.
b. Suppose that the assumption(s) made in part a was not justifiable. A researcher decided to take a random sample of 81 full professors from the Canadian University system.
i. What is the sampling distribution of the sample mean ¯ x ? Explain.
ii. Find the mean and standard deviation of the sampling distribution of ¯ x.
iii. What is the probability that the sample average of this sample is less than 130? Compare your answer with the answer given in a. Summarize your findings.
iv. Will the probability in (iii) change if you change the wording from “less than” to “less than or equal to”? Why or why not?
v. Can you compute the probability that the sample average IQ is exactly 135? Justify your answer.
vi. If the sample mean ¯ x is actually actually 130, what can be said about the claim that µ = 140.
vii. What is the probability that the sample mean differs from the population mean by more than 2?
viii. Within what limits do you expect the sample average to be
with the probability 0.95?
In: Statistics and Probability
Two television stations compete with each other for viewing audience. Local programming options for the 5:00 P.M. weekday time slot include a sitcom rerun, an early news program, or a home improvement show. Each station has the same programming options and must make its preseason program selection before knowing what the other television station will do. The viewing audience gains in thousands of viewers for Station A are shown in the payoff table.
| Station B | ||||
| Sitcom Rerun |
News Program |
Home Improvement |
||
| b1 | b2 | b3 | ||
| Station A | Sitcom Rerun a1 | 10 | -5 | 3 |
| News Program a2 | 8 | 7 | 6 | |
| Home Improvement a3 | 4 | 8 | 7 | |
Determine the optimal strategy for each station. Round your answers to two decimal places. If your answer is zero, enter zero "0".
The optimal strategy is for Station A to implement:
strategy a1 with probability ?
strategy a2 with probability ?
strategy a3 with probability ?
The optimal strategy is for Station B to implement:
strategy b1 with probability ?
strategy b2 with probability ?
strategy b3 with probability ?
What is the value of the game? Round your answer to two decimal places.
The value of the game =
Note:Please show how to solve for probability value for a1, a2, a3, b1, b2, b3, and the value of the game.
In: Statistics and Probability
A department store sells sport shirts in three sizes (small, medium, and large), three patterns (plaid, print, and stripe), and two sleeve lengths (long and short). The accompanying tables give the proportions of shirts sold in the various category combinations.
| Short-sleeved | |||
|---|---|---|---|
| Pattern | |||
| Size | Pl | Pr | St |
| S | 0.04 | 0.02 | 0.05 |
| M | 0.09 | 0.05 | 0.12 |
| L | 0.03 | 0.07 | 0.08 |
| Long-sleeved | |||
|---|---|---|---|
| Pattern | |||
| Size | Pl | Pr | St |
| S | 0.03 | 0.02 | 0.03 |
| M | 0.08 | 0.08 | 0.07 |
| L | 0.04 | 0.02 | 0.08 |
(a) What is the probability that the next shirt sold is a
medium, long-sleeved, print shirt?
(b) What is the probability that the next shirt sold is a medium
print shirt?
(c) What is the probability that the next shirt sold is a
short-sleeved shirt? A long-sleeved shirt?
| short-sleeved shirt | ||
| long-sleeved shirt |
(d) What is the probability that the size of the next shirt sold is
medium?
What is the probability that the pattern of the next shirt sold is
a print?
(e) Given that the shirt just sold was a short-sleeved plaid, what
is the probability that its size was medium? (Round your answer to
three decimal places.)
(f) Given that the shirt just sold was a medium plaid, what is the
probability that it was short-sleeved? Long-sleeved? (Round your
answer to three decimal places.)
| short-sleeved | ||
| long-sleeved |
In: Statistics and Probability
Let x be a random variable that represents white blood cell count per cubic milliliter of whole blood. Assume that x has a distribution that is approximately normal, with mean μ = 6700 and estimated standard deviation σ = 2200. A test result of x < 3500 is an indication of leukopenia. This indicates bone marrow depression that may be the result of a viral infection.
(a) What is the probability that, on a single test, x
is less than 3500? (Round your answer to four decimal
places.)
(b) Suppose a doctor uses the average x for two tests
taken about a week apart. What can we say about the probability
distribution of x?
a.The probability distribution of x is not normal.
b.The probability distribution of x is approximately normal with μx = 6700 and σx = 1555.63.
c.The probability distribution of x is approximately normal with μx = 6700 and σx = 1100.00.
d.The probability distribution of x is approximately normal with μx = 6700 and σx = 2200.
What is the probability of x < 3500? (Round your answer
to four decimal places.)
(c) Repeat part (b) for n = 3 tests taken a week apart.
(Round your answer to four decimal places.)
(d) Compare your answers to parts (a), (b), and (c). How did the
probabilities change as n increased?
a.The probabilities increased as n increased.
b.The probabilities stayed the same as n increased.
c.The probabilities decreased as n increased.
In: Statistics and Probability
In a survey of a group of men, the heights in the 20-29 age group were normally distributed, with a mean of 67.1 inches and a standard deviation of 3.0 inches. A study participant is randomly selected. Complete parts (a) through (d) below.
(a) Find the probability that a study participant has a height that is less than 68 inches. The probability that the study participant selected at random is less than 68 inches tall is ___ (round to 4 decimal points)
(b) Find the probability that a study participant has a height that is between 68-72 inches. The probability that the study participant selected at random is between 68-72 inches is __ (round to 4 decimal points)
(c) Find the probability that a study participant has a height that is more than 72 inches. The probability that the study participant selected at random is more than 72 inches is __ (round to 4 decimal points)
(d) Identify any unusual events. Explain your reasoning. Choose the correct answer below.
A. The event in part (a) is unusual because its probability is less than 0.05.
B. There are no unusual events because all probabilities are greater than 0.05
C. The events in parts (a), (b), and (c) are unusual because all of the probabilities are less than 0.05
D. The events in parts (a) and (c) are unusual because its probabilities are less than 0.05
In: Statistics and Probability
Let x be a random variable that represents the level of glucose in the blood (milligrams per deciliter of blood) after a 12 hour fast. Assume that for people under 50 years old, x has a distribution that is approximately normal, with mean μ = 75 and estimated standard deviation σ = 50. A test result x < 40 is an indication of severe excess insulin, and medication is usually prescribed.
(a) What is the probability that, on a single test, x
< 40? (Round your answer to four decimal places.)
(b) Suppose a doctor uses the average x for two tests
taken about a week apart. What can we say about the probability
distribution of x? Hint: See Theorem 6.1.
The probability distribution of x is approximately normal with μx = 75 and σx = 25.00.The probability distribution of x is approximately normal with μx = 75 and σx = 35.36. The probability distribution of x is not normal.The probability distribution of x is approximately normal with μx = 75 and σx = 50.
What is the probability that x < 40? (Round your answer
to four decimal places.)
(c) Repeat part (b) for n = 3 tests taken a week apart.
(Round your answer to four decimal places.)
(d) Repeat part (b) for n = 5 tests taken a week apart.
(Round your answer to four decimal places.)
In: Statistics and Probability