Cause and Effect T14

On Your OwnU

nit 1

Applications

1 a. Jumper weight is the most plausible independent variable and stretched cord length the dependent variable because it changes as an effect of changed jumper weight.

b.

c. Reasonable estimates are:

i. 47 feet ii. 57 feet iii. 75 feet

d. In this situation, it probably does make sense to connect the dots to indicate stretched cord length for weight values between the given data points because there is a clear trend and no reason to think the relationship would be any different for weight values between those in the table. Also, values between those in the table make sense (i.e., jumper weight is a continuous variable).

e. Stretched cord length increases steadily as jumper weight increases. For every 25-pound increase in jumper weight, there is a 5-foot increase in stretched cord length or about 0.2 feet per pound of weight.

f. The given rule does produce the (jumper weight, stretched cord length) values in the data table. For example, when w = 100, the rule says L = 50, and when w = 200, the rule says L = 70.

2 a. In this case, price per jump is the natural independent variable and number of customers is the natural dependent variable because it is affected by the price charged.

b.

On Your Own

Cause and Effect T15

INSTRUCTIONAL NOTE Problem 2 provides an opportunity to discuss the issue of rounding to whole numbers of customers.

Unit 1

c. In this situation, connecting the dots is a reasonable way to visualize the pattern and to make estimates of numbers of customers for prices between those for which the given data provide direct evidence. However, since the pattern does seem linear, it may be more reasonable to draw a single line that matches the pattern well.

d. Interpolating and extrapolating in a roughly linear pattern from the given data points, expected number of customers at the suggested prices would be about:

i. 23 ii. 19 iii. 10

e. The overall pattern of change is one in which number of customers declines as price per jump increases. The pattern is roughly linear, with on average a drop of about 3–4 customers for every $5 increase in price.

f. The values produced by the rule are quite close to the actual data.

Price per Jump (in dollars) 15 20 25 28 30

Number of Customers 24.5 21 17.5 15.4 14

3 a. Price per Jump (in dollars) 15 20 25 28 30

Daily Income (in dollars) 375 440 450 420 420

b. In this case, the natural independent variable is price per jump while the natural dependent variable is daily income because the income is based on the price charged for jumps.

c. A plot of the (price per jump, daily income) data will look like the graph to the right:

d. It would make some sense to connect the points since prices between those for which there are income estimates do make sense, and it is reasonable to believe that the incomes associated with those prices would be in the same pattern as that of the data points already known. However, since the pattern does not seem linear, it may be more reasonable to draw a smooth curve through the points.

e. The overall pattern appears to suggest an increase in income as prices approach the $20–$25 range, and then a decline as the price rises higher.

f. A jump price somewhere around $24 or $25 seems to yield the maximum daily income.

4 a. i. Since the probability of two heads or two tails is the same as the probability of one head and one tail (each are 1 � 2 ), the long-run cumulative winnings or losses from this game should be about 0.

ii. The graph would be points oscillating slightly above and slightly below the x-axis.

b. Answers will vary, but should be roughly in the pattern described in Part a.

c. Explanations will depend on the match between predicted and actual results.

On Your Own

Cause and Effect T16

Unit 1

5 a.

b. i. 1.5 ounces would cost $0.63.

ii. 4.25 ounces would cost $1.35.

iii. 7 ounces cost $1.83, assuming the pattern of increasing the cost by 24¢ for each ounce continues.

c. One possible suggestion would be as a step function as seen in the graph below.

6 a.

b. Race time decreases rapidly as speed increases from 2 meters per second, but for higher speeds a comparable increase in speed does not produce as significant a reduction in time.

c. t = 400 � s

d. The change from 2 m/sec to 4 m/sec produces the greater reduction in race time. That is shown by the fact that the graph drops more rapidly for increases in speed at lower speeds.

7 a.

Average Speed (in m/sec) 2 3 4 5 6 7 8 9 10

Race Time (in sec) 200 133 100 80 67 57 50 44 40

Race Time (in sec) 40 50 60 70 80 90 100 110 120

Average Speed (in m/sec) 2.5 2.0 1.7 1.4 1.3 1.1 1.0 0.9 0.8

On Your Own

Cause and Effect T17

Unit 1

b. Race speed decreases rapidly as time increases from 40 seconds, but for higher times the increase in time does not produce as significant a reduction in speed.

c. s = 100 � t

d. The change from 50 seconds to 60 seconds produces the greater reduction in race speed. That is shown by the fact that the graph drops more rapidly for increases in time at lower times.

8 a–e. Answers to each part of this task will vary, but there ought to be a general trend for increase in temperature to cause increase in number of customers (except perhaps when it begins to get too hot).

f. The projected income should be obtained by multiplying the number of customers by $15 in each case.

g. With very low or very high temperatures, the patterns set at more reasonable swimming temperatures might have little relevance, since almost no one would go swimming in 40°F weather or even in 110°F weather.

On Your Own

Cause and Effect T18

Unit 1

Connections

9 a. The scatterplot shows some trend for higher latitude cities to have lower temperatures. However, there is little midsummer variation due to latitude among cities in the lower latitudes.

b. The scatterplot shows a fairly consistent trend for higher latitude cities to have lower midwinter temperatures.

c. Among the factors students may include are those that deal with geographic differences such as distance above sea level and location near or far from large bodies of water like oceans.

On Your Own

Cause and Effect T19

Unit 1

10 a. There are two possible equally likely outcomes, “heads” or “tails,” so

the probability is 1 � 2 .

b. There are four possible equally likely outcomes: HH, HT, TH, or TT. So

the probability of getting two heads is 1 � 4 .

c. A tree diagram will show 8 possibilities, one of which is three heads.

Thus, the probability of getting three heads is 1 � 8 .

d. A tree diagram will show 16 possibilities, one of which is four heads.

Thus, the probability of getting four heads is 1 � 16 .

e. As the number of coins increases by one, the probability of getting all

heads decreases by a factor of 0.5. (The rule p = 1 � 2n , where p is the

probability of getting all heads and n is the number of coin flips, could also be used to describe the situation.)

11 a. 7 + 7 + 9 � 3 ≈ 7.67

b. 8 · 7 + 9 � 9 ≈ 7.22

c. With only three scores, the 9 score has more of an “influence” on the average. In other words, in Part a, the 9 is a higher percentage of the total points earned (23) than in Part b (65). (One way to illustrate this idea is to picture 1,000 quizzes with one 9 and 999 7s. This 9 has even less influence than the one in Part b, and the average score continues to get closer to 7.)

12 a. (1) Enrollment increased by 250 students.

(2) by 25%

(3) by an average of 50 students per year

b. (1) The balance increased by $75.

(2) by 50%

(3) by an average of $25 per month

On Your Own

Cause and Effect T20

Unit 1

c. (1) The number of drinks decreased by 60.

(2) by 30%

(3) by an average of 7.5 drinks per hour

d. (1) Mike’s time decreased by 30 seconds.

(2) by 20%

(3) by an average of 7.5 seconds per month

13 Answers will vary, depending on examples of situations that students choose to describe.

Reflections

14 There should be some margin of error, especially on the safe side.