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Date:
Block:
AP Statistics – Chapter 7 – Random Variables – Means & Variances – Pete’s Jeep Tours Activity
Linear Transformations
Pete’s jeep Tours offers a popular half-day trip in a tourist area.
There must be at least 2 passengers for the trip to run, and the
vehicle will hold up to 6 passengers. The number of passengers X
on a randomly selected day has the following probability
distribution.
Passengers Xi
2
3
4
5
6
Probability pi
0.15
0.25
0.35
0.20
0.05
1. Using the histogram provided, estimate the mean and standard deviation.
2. Calculate and interpret the mean, variance, and standard deviation.
 2X 
X =
X =
Pete charges $150 per passenger. Let C = the total amount of
money that Pete colle3ct on a randomly selected trip. Because of
the amount of money Peter collects from the trip is just $150
times the number of passengers, we can write C = 150X. From the
probability distribution of X, we can see that the chance of having
two people (X = 2) on the trip is 0.15. In that case, ($150)(2) =
$300. So one possible value of C is $300, and its corresponding
probability is 0.15.
Passengers Ci
300
Probability pi
0.15
0.25
0.35
0.20
0.05
3. Complete the probability distribution for C = the total amount of money collected.
4. Calculate and interpret the mean and standard deviation.
C =
 2C 
C =
5. What happens to the shape, center, and spread of the distribution?
6. Check the following to see if they are true. What “rules” can you determine about multiplying a random
 C  150 X
 C  150 C
variable by a constant.
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It costs Pete $100 to buy permits, gas, and ferry pass for each halfday trip. The amount of profit V that Pete makes from the trip is
the total amount of money C that he collects from passengers
minus $100. That is, V = C – 100. If Pete has only two passengers
on the trip (X = 2), then C = 300 and V = 200. From the probability
distribution C, the chance that this happens is 0.15. So the smallest
possible value of V is $200; its corresponding probability is 0.15.
Passengers Vi
200
Probability pi
0.15
0.25
0.35
0.20
0.05
7. Complete the probability distribution for V = the total amount of profit.
8. Calculate and interpret the mean and standard deviation.
 V2 
V =
V =
9. What happens to the shape, center, and spread of the distribution?
10. Check the following to see if they are true. What “rules” can you determine about multiplying a random
V   C  100
 V   C  100
variable by a constant.
Pete’s sister Erin, who lives near a tourist area in another part of
the country, is impressed by the success of Pete’s business. She
decides to join the business, running tours on the same days as
Pete in her slightly smaller vehicle, under the name Erin’s
Adventures. After a year of steady bookings, Erin discovers that
the number of passengers Y on her half-day tours has the
following probability distribution.
Passengers Yi
2
3
4
5
Probability pi
0.3
0.4
0.2
0.1
11. Calculate and interpret the mean, variance, and standard deviation.
Y 
 Y2 
Y 
In this section, we learned that…
1. Adding a constant a (which could be negative) to a random variable increases (or decreases) the mean of the
random variable by a, but does not affect its standard deviation or the shape of its probability distribution.
2. Multiplying a random variable by a constant b (which could be negative) multiplies the mean of the random
variable by b and the standard deviation by |b| but does not change the shape of its probability distribution.
3. A linear transformation of a random variable involves adding a constant a, multiplying by a constant b, or
both. If we write the linear transformation of X in the form Y = a + bX, the following about are true about Y:
Shape: same as the probability distribution of X. Center: µY = a + bµX
Spread: σY = |b|σX
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Combing Random Variables
Let X = number of passengers on Pete’s trip and Y = number of
passengers on Erin’s Trip. Let T = X + Y, as before. How many total
passengers T can Pete and Erin expect to have on their tours on a
randomly selected day?
How much variability is there in the total number of passengers
who go on Pete’s and Erin’s tours on a randomly chosen day? Does
the variability increase, decrease, or stay the same? Let’s think
about the possible values of T = X + Y. The number of passengers X
on Pete’s tour is between 2 and 6 (range of 4), and the number of
passengers on Erin’s tour is between 2 and 5 (range of 3). So the
total number of passengers T is between 4 and 11 (range of 7). Is it
true that RangeT  Range X  RangeY ?
That is, there is more variability in the values of T than in the
values of X or Y alone. This makes sense, because the variation in
X and the variation in Y both contribute to the variation in T.
The only way to determine the probability
for any value of T is if X and Y are
independent random variables.
Definition: If knowing whether any event
involving X alone has occurred tells us
nothing about the occurrence of any event
involving Y alone, and vice versa, then X
and Y are independent random variables.
Probability models often assume
independence when the random variables
describe outcomes that appear unrelated
to each other. You should always ask
whether the assumption of independence
seems reasonable. In our investigation, it is
reasonable to assume X and Y are
independent since the siblings operate
their tours in different parts of the country.
Let T = X + Y. Consider all possible combinations of the values of X
and Y. Calculate the probability distribution for T.
4
Passengers Ti
Probability pi
5
6
7
8
9
10
11
0.045 0.135 0.235 0.265 0.190 0.095 0.030 0.005
12. Calculate and interpret the mean, variance, and standard
deviation.
T 
13. Does  T   X   Y ?
 2T 
T 
Does  2T   2X   Y2 ?
Does  T   X   Y ?
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In
this section, we learned that…
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 If X and Y are any two random
variables, then
 X Y   X   Y .
 If X and Y are independent random
variables, then
2
2
2
X Y
X
Y

  
 The sum or difference of
independent Normal random
variables follows a Normal
distribution.
14. Earlier, we defined X = the number of passengers on Pete’s trip, Y = the number of passengers on Erin’s
trip, C = the amount of money that Pete collects on a randomly selected day, and G = the amount of money
that Erin collects on a randomly selected day. We also found the means and standard deviations of these
variables:
 X  3.75
 X  1090
.
 Y  310
.
 Y  0.943
 C  562.50
 C  16350
.
 G  542.50
 G  165.03
a. Erin charges $175 per passenger for her trip. Let G = the amount of money that she collects on a
randomly selected day. Find the mean and standard deviation of G.
b. Calculate the mean and the standard deviation of the total amount that Pete and Erin collect on a
randomly selected day.
c. Calculate the mean and standard deviation of the differences D = C – G in the amounts that Pete and
Erin collect on a randomly selected day