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Suppose the fill amount of bottles of soft drink has been found to be normally distributed with a mean of 2.0 liters and a standard
deviation of 0.08 liters. Bottles that contain less than 95% of the listed net content can make the manufacturer subject to penalty by
the state office of consumer affairs, whereas bottles that have a net content above 2.10 liters may cause excess spillage upon opening.
Let x denote the amount of fill in a randomly selected bottle of soft drink.
a. Determine the mean and standard deviation of the random variable x.
b. Find P(1.85 # x # 2.00)
c. Determine P(x > 2.05)
d. Determine P(x < 1.90 or x > 2.10)
e. Determine P(x < 1.95)
f. Using the empirical rule, the probability is 0.9544 that a bottle of soft drink will have a fill amount between ______ and ______
liters.
g. Find the standardized version, z, of the random variable x.
h. What is the probability distribution of the random variable z?
i. What does the random variable z represent?
j. How many standard deviations from the mean is a fill amount of 1.935L? 2.2L? 2.0L?
The tread life of a particular brand of tire is a random variable best described by a normal distribution with a mean of 60,000
miles and a standard deviation of 8,500 miles.
a. If the manufacturer guarantees the tread life for the first 45,000 miles, what proportion of the tires will need to be replaced under
warranty?
b. If the manufacturer guarantees the tread life for the first 40,000 miles, what proportion of the tires will need to be replaced under
warranty?
7. The distribution of the demand (in number of units per unit time) for a product can often be approximated by a normal probability
distribution. For example, a bakery has determined that the number of loaves of its white bread demanded daily has a normal
distribution with mean 7,200 loaves and standard deviation 350 loaves. Based on cost considerations, the company has decided that
its best strategy is to produce a sufficient number of loaves so that it will fully supply demand on 94% of all days. (Ten points)
a. How many loaves of bread should the company produce?
b. Based on the production in part a, on what percentage of days will the company be left with more than 500
loaves of unsold bread?
In a Time/CNN poll of 520 employed adults 60% said that companies today are less loyal to their employees than 10 years ago.
Assuming that this percentage is true for the current population of all employed adults, find the probabilities that in a random
sample of 200 such adults, the number who hold this view is
a. 105 to 115
b. more than 128.
c. exactly 110.
Fuel efficiency for cars made in the USA is normally distributed with a mean of 28.8 mpg and a standard deviation of 8.2 mpg.
A special tax is planned for the worst 5% of cars made in the USA in terms of fuel efficiency. How low must the fuel efficiency be in
order to require this special tax?
The following table gives the (discrete) probability distribution of television sets sold on a given day at an electronics store.
x 0 1 2 3 4 5 6
P(x) 0.05 0.12 0.25 0.28 0.16 0.10 0.04
Calculate the mean and standard deviation of x. Comment on the interpretation the mean value.
New Horizons Airlines wants to estimate the mean number of unoccupied seats per flight to Germany over the past year. New
Horizons is looking for increased business with Germany. To accomplish the investigation, records of 400 flights are randomly
selected from the files, and the number of unoccupied seats is noted for each flight. The sample mean and standard deviation are 8.1
seats and 5.7 seats, respectively. (Ten points)
a. Compute the 95% confidence interval for population mean unoccupied seats per flight. Interpret this interval.
b. How many flights should be sampled so that we may construct a 95% confidence interval within 1 seat of the population mean.
Assume the sample size will be larger than 30.
Suppose the population mean, :, is four times larger than the population standard deviation, F, of a particular population. You do
know that P(x # 5) is equal to 0.898. Compute the population mean and variance.
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