Compute the following.
(a) P9, 4.
(b) C9, 4.
(c) P4, 4.
(d) C5, 5.
The student government claims that 75% of all students favor an increase in student fees to buy indoor potted plants for the
classrooms. A random sample of 9 students produced 2 in favor of the project.
(a) What is the probability that 2 or fewer in the sample will favor the project, assuming the student government's claim is
correct? (Use 3 decimal places.)
(b) Do the the data support the student government's claim, or does it seem that the percentage favoring the increase in fees is
less than 75%?
The data seem to indicate that the percent favoring the increase in fees is equal to 75%.
The data seem to indicate that the percent favoring the increase in fees is less than 75%.
The data do not give us any indication that the percent favoring the increase in fees differs from 75%.
The data seem to indicate that the percent favoring the increase in fees is greater than 75%.
It is estimated that 75% of a grapefruit crop is good; the other 25% have rotten centers that cannot be detected unless the
grapefruit are cut open. The grapefruit are sold in sacks of 9. Let r be the number of good grapefruit in a sack.
(a) Make a histogram of the probability distribution of r.
(b) What is the probability of getting no more than one bad grapefruit in a sack? (Round your answer to three decimal places.)
0.158
What is the probability of getting at least one good grapefruit? (Round your answer to three decimal places.)
(c) What is the expected number of good grapefruit in a sack? (Round your answer to one decimal place.)
grapefruit
(d) What is the standard deviation of the r probability distribution? (Round your answer to two decimal places.)
grapefruit
Express Courier Service has found that the delivery time for packages is normally distributed, with mean 15 hours and standard
deviation 3 hours.
(a) For a package selected at random, what is the probability that it will be delivered in 18 hours or less? (Round your answer to
four decimal places.)
(b) What should be the guaranteed delivery time on all packages in order to be 90% sure that the package will be delivered
before this time? (Hint: Note that 10% of the packages will be delivered at a time beyond the guaranteed time period.) (Round
your answer to one decimal place.)
hr
A large tank of fish from a hatchery is being delivered to a lake. The hatchery claims that the mean length of fish in the tank is
15 inches, and the standard deviation is 4 inches. A random sample of 36 fish is taken from the tank. Let x be the mean sample
length of these fish. What is the probability that x is within 0.5 inch of the claimed population mean? (Round your answer to
four decimal places.)
Future Electronics makes compact disc players. Its research department found that the life of the laser beam device is normally
distributed, with mean 4620 hours and standard deviation 490 hours.
(a) Find the probability that the laser beam device will wear out in 5000 hours or less. (Round your answer to four decimal
places.)
(b) Future Electronics wants to place a guarantee on the players so that no more than 10% fail during the guarantee period.
Because the laser pickup is the part most likely to wear out first, the guarantee period will be based on the life of the laser
beam device. How many playing hours should the guarantee cover? (Round your answer down to the nearest hour.)
hr
Assume that IQ scores are normally distributed, with a standard deviation of 17 points and a mean of 100 points. If 70 people
are chosen at random, what is the probability that the sample mean of IQ scores will not differ from the population mean by
more than 2 points? (Round your answer to four decimal places.)
What happens if we want several confidence intervals to hold at the same time (concurrently)? Do we still have the same level
of confidence we had for each individual interval?
(a) Suppose we have two independent random variables x1 and x2 with respective population means μ1 and μ2. Let us say that
we use sample data to construct two 80% confidence intervals.
Confidence Interval
A1 < μ1 < B1
A2 < μ2 < B2
Confidence Level
0.80
0.80
Now, what is the probability that both intervals hold together? Use methods of Section 5.2 to show that
P(A1 < μ1 < B1 and A2 < μ2 < B2) = 0.64.
Hint: We are combining independent events.
P(A1 < μ1 < B1 and A2 < μ2 < B2) = P(A1 < μ1 < B1) – P(A2 < μ1 < B2)
P(A1 < μ1 < B1 and A2 < μ2 < B2) = P(A1 < μ1 < B1) + P(A2 < μ1 < B2)
P(A1 < μ1 < B1 and A2 < μ2 < B2) = P(A1 < μ1 < B1) · P(A2 < μ1 < B2)
P(A1 < μ1 < B1 and A2 < μ2 < B2) = P(A1 < μ1 < B1) / P(A2 < μ1 < B2)
If the confidence is 64% that both intervals hold together, explain why the risk that at least one interval does not hold (i.e., fails)
must be 36%.
P(at least one interval fails to capture μi) = P(both intervals capture their μi) + 1
P(at least one interval fails to capture μi) = 1 – P(both intervals capture their μi)
P(at least one interval fails to capture μi) = P(both intervals capture their μi) – 1
P(at least one interval fails to capture μi) = 1 + P(both intervals capture their μi)
(b) Suppose we want both intervals to hold with 75% confidence (i.e., only 25% risk level). How much confidence c should each
interval have to achieve this combined level of confidence? (Assume that each interval has the same confidence level c.)
Hint: P(A1 < μ1 < B1 and A2 < μ2 < B2) = 0.75
P(A1 < μ1 < B1 ✕ A2 < μ2 < B2) = 0.75
c ✕ c = 0.75
Now solve for c. (Use 3 decimal places.)
(c) If we want both intervals to hold at the 90% level of confidence, then the individual intervals must hold at a higher level of
confidence. Write a brief but detailed explanation of how this could be of importance in a large, complex engineering design
such as a rocket booster or a spacecraft.
Three-circle, red-on-white is one distinctive pattern painted on ceramic vessels of the Anasazi period found at an archaeological
site. At one excavation, a sample of 167 potsherds indicated that 66 were of the three-circle, red-on-white pattern.
(a) Find a point estimate p̂ for the proportion of all ceramic potsherds at this site that are of the three-circle, red-on-white
pattern. (Round your answer to four decimal places.)
(b) Compute a 95% confidence interval for the population proportion p of all ceramic potsherds with this distinctive pattern
found at the site. (Round your answers to three decimal places.)
lower limit
upper limit
A research group conducted an extensive survey of 3122 wage and salaried workers on issues ranging from relationships with
their bosses to household chores. The data were gathered through hour-long telephone interviews with a nationally
representative sample. In response to the question, "What does success mean to you?" 1497 responded, "Personal satisfaction
from doing a good job." Let p be the population proportion of all wage and salaried workers who would respond the same way
to the stated question. How large a sample is needed if we wish to be 95% confident that the sample percentage of those
equating success with personal satisfaction is within 2.6% of the population percentage? (Hint: Use p ≈ 0.48 as a preliminary
estimate. Round your answer up to the nearest whole number.)
workers
A research group conducted an extensive survey of 2955 wage and salaried workers on issues ranging from relationships with
their bosses to household chores. The data were gathered through hour-long telephone interviews with a nationally
representative sample. In response to the question, "What does success mean to you?" 1478 responded, "Personal satisfaction
from doing a good job." Let p be the population proportion of all wage and salaried workers who would respond the same way
to the stated question. Find a 90% confidence interval for p. (Round your answers to three decimal places.)
lower limit
upper limit
A machine in the student lounge dispenses coffee. The average cup of coffee is supposed to contain 7.0 ounces. A random
sample of eight cups of coffee from this machine show the average content to be 7.26 ounces with a standard deviation
of 0.70 ounce. Do you think that the machine has slipped out of adjustment and that the average amount of coffee per cup is
different from 7 ounces? Use a 5% level of significance.
(a) What is the level of significance?
(b) What is the value of the sample test statistic? (Round your answer to three decimal places.)
(c) Find the P-value. (Round your answer to four decimal places.)
Sketch the sampling distribution and show the area corresponding to the P-value.
(c)
(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data
statistically significant at level α?
At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.
At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.
At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
A hospital reported that the normal death rate for patients with extensive burns (more than 40% of skin area) has been
significantly reduced by the use of new fluid plasma compresses. Before the new treatment, the mortality rate for extensive
burn patients was about 60%. Using the new compresses, the hospital found that only 41 of 88 patients with extensive burns
died. Use a 1% level of significance to test the claim that the mortality rate has dropped.
(a) What is the level of significance?
What is the value of the sample test statistic? (Round your answer to two decimal places.)
(c) Find the P-value. (Round your answer to four decimal places.)
Sketch the sampling distribution and show the area corresponding to the P-value.
(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically
significant at level α?
At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.
At the α = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.
At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the α = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.