Download Worksheet on Linear Transformations from Class

AP Stats
Linear Problems
Name___________________
1. Use the following distribution to answer the questions below.
x
4
6
8
10
p(x)
0.22
0.31
0.12
0.35
y
20
30
40
50
p(y)
.54
.22
.14
.1
a. Find
4 x  y
b. Find
 4 x y
c. Find
5 x  6
d. Find
 5 x6
2. Suppose you earn $12 an hour for tutoring but spend $8 an hour for dance lessons. You save the difference
between what you earn and the cost of your lessons. The number of hours you spend on each activity in a
week varies independently according to the probability distributions shown below. Find your expected
weekly savings and the standard deviation of your weekly savings.
x = # Dance
hours
0
1
2
y=#Tutoring
Hours
1
2
3
4
p(x)
0.4
0.3
0.3
p(y)
0.3
0.3
0.2
0.2
3. You can’t decide whether to buy Brand A or Brand B as your new dishwasher. Each brand is expected to last
about 10 years. Brand A costs $950 with an unlimited number of repairs at $150 each. Brand B cost more
($1200) but comes with an unlimited number of free repairs. Which dishwasher you buy depends on the
number of repairs you expect Brand A to require. You investigate and find the information shown below.
Find the expected cost of each brand. What is the advantage of buying Brand A? Brand B?
x = # repairs
p(x)
0
0.4
1
0.3
2
0.15
3
0.1
4
0.05
4. Suppose you sell box lunches. One box contains ham sandwiches and the other has turkey. The probability
distribution for the number of boxes sold each day are shown below. If you make $5 profit for each ham
box and $4.50 for each turkey box, find the expected profit and deviation in profit for each day.
H (#ham)
10
15
20
25
30
p(H)
0.2
0.23
0.25
0.17
0.15
T (#turkey)
5
10
15
20
25
p(T)
0.16
0.19
0.26
0.24
0.15
5. A small ferry runs every half hour from one side of a large river to the other. The number of cars, x, on a
randomly chosen ferry trip has the probability distribution shown below. The cost for the ferry trip is $5.
The ferry company’s expenses are $20 per trip. Define the random variable Y to be the amount of profit
made by the ferry company on a randomly selected trip.
1st Ferry
2nd Ferry – further upriver
Cars:
0
1
2
3
4
5
Prob: 0.02 0.05 0.08 0.16 0.27 0.42
Cars:
0
1
2
3
4
5
6
Prob: 0.01 0.03 0.04 0.12 0.18 0.27 0.35
a. What is the mean profit per trip and find the typical deviation in this profit.
b. This company runs a second ferry further up the river. The distribution for the number of cars is shown
above and to the right. This ferry costs $6 and the company’s expenses are $22 per trip. What is the
mean profit and typical deviation for the running of this ferry?
c. What is the mean and standard deviation for the total profit that the company makes from these two
ferries?
6. Hannover high School has the best women’s swimming team in the region. The 400-meter freestyle relay
team is undefeated this year. In this race, each swimmer swims 100 meters. The times, in seconds, for the
four swimmers this season are approximately Normally distributed with means and standard deviations as
shown below. Assuming that the swimmer’s individual times are independent, find the probability that the
total team time in the 400-meter freestyle relay is less than 220 seconds.
Swimmer
Wendy
Jill
Carmen
Latrice
Mean
55.2
58
56.3
54.7
Std.
Dev.
2.8
3
2.6
2.7