The US Environmental Protection Agency reports the average fuel costs for various makes of
cars. In recent years the fuel costs for mid‐size Japanese cars averaged $1200. Suppose that the
annual fuel costs for various makes of automobiles are normally distributed and that the
standard deviation of annual fuel costs for the mid‐size Japanese cars is $200. A mid‐size car
from the specified year is randomly selected from those that were sold. Answer the questions
below. It is strongly recommended that you do a sketch in each case to show the area(s) or
probability you are evaluating.
25. What is the probability that the annual fuel cost for a randomly selected car is greater
than $1000?
(a) 0.9323
(b) 0.8255
(c) 0.3413
(d) 0.8413
(e) 0.2823
26. What is the probability that the annual fuel cost is less than $1050?
(a) 0.2946
(b) 0.2734
(c) 0.7734
(d) 0.2266
(e) 0.0823
27. What is the probability that the annual fuel cost is between $1100 and $1350?
(a) 0.4322
(b) 0.1915
(c) 0.2417
(d) 0.6247
(e) 0.4649
28. What is the probability that the annual fuel cost is $1500 or less?
(a) 0.7088
(b) 0.4332
(c) 0.0668
(d) 0.8332
(e) 0.9332
29. What is the probability that the annual fuel cost is $1250 or more?
(a) 0.0987
(b) 0.4013
(c) 0.9010
(d) 0.9013
(e) 0.5261
30. What is the probability that the annual fuel cost is between $1225and $1275?
(a) 0.0908
(b) 0.0517
(c) 0.1480
(d) 0.0963
(e) 0.4261
31. Suppose that the standard deviation of the fuel cost is unknown. If 80% of all mid‐size
cars of that year had an annual fuel cost greater than $1030, what would be the value of
standard deviation if the mean fuel cost is $1200?
(a) 231.70
(b) 200.00
(c) 202.38
(d) 230.00
(e) 235.09
32. Suppose that the average annual fuel cost is unknown. What is the value of the mean
annual fuel cost for the mid‐size cars if 30.5% of the annual fuel costs are less than $1100
when the standard deviation is $200.
(a) 1100.70
(b) 1205.00
(c) 1202.00
(d) 1788.00
(e) 1235.09
DID YOU FIND THE ANSWER FOR THESE QUESTIONS?
To calculate the probabilities, we will use the standard normal distribution and the given mean and standard deviation. The standard normal distribution has a mean of 0 and a standard deviation of 1.
To solve the problems, we will convert the given values to z-scores using the formula:
z = (x - μ) / σ
where:
- z is the z-score
- x is the value we want to find the probability for
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
Once we have the z-score, we can use the standard normal distribution table (also known as the Z-table) or a calculator to find the corresponding probability.
Let's solve the problems one by one:
25. What is the probability that the annual fuel cost for a randomly selected car is greater than $1000?
First, we calculate the z-score:
z = ($1000 - $1200) / $200
z = -1
Using the standard normal table, we find that the probability corresponding to a z-score of -1 is 0.1587.
However, we want the probability of the fuel cost being greater than $1000, so we subtract 0.1587 from 1:
P(x > $1000) = 1 - 0.1587 = 0.8413
Hence, the answer is (d) 0.8413.
26. What is the probability that the annual fuel cost is less than $1050?
First, we calculate the z-score:
z = ($1050 - $1200) / $200
z = -0.75
Using the standard normal table, we find that the probability corresponding to a z-score of -0.75 is 0.2266.
Hence, the answer is (d) 0.2266.
27. What is the probability that the annual fuel cost is between $1100 and $1350?
First, we calculate the z-scores:
z1 = ($1100 - $1200) / $200
z1 = -0.5
z2 = ($1350 - $1200) / $200
z2 = 0.75
Using the standard normal table, we find that the probability corresponding to a z-score of -0.5 is 0.1915, and the probability corresponding to a z-score of 0.75 is 0.7734.
To find the probability between these two values, we subtract the probability corresponding to the lower z-score from the probability corresponding to the higher z-score:
P($1100 < x < $1350) = 0.7734 - 0.1915 = 0.5819
Hence, the answer is not provided.