Problem-Solving and Data Analysis Practice Questions — SAT — Page 7

Question 121Question →

A specialized laser cutter cuts steel sheet metal at a constant rate of $18$ inches per minute. At this rate, how many seconds does it take the laser cutter to cut a length of $6$ feet of steel? ( $1\text{ foot} = 12\text{ inches}$ )

Show answer & explanation

Answer: 240

Answer

The correct answer is 240.

To find the number of seconds it takes the laser cutter to cut

6

feet of steel, we first convert the length to inches:

6\text{ feet} \times 12\text{ inches/foot} = 72\text{ inches}

. Next, we divide this length by the cutting rate to find the duration in minutes:

72\text{ inches} \div 18\text{ inches/minute} = 4\text{ minutes}

. Finally, we convert the minutes into seconds: $4\text{ minutes} \times 60\text{ seconds/minute} = 240\text{ seconds}.

Step-by-Step Solution

1

Convert the total cutting distance from feet to inches.

72\text{ inches}

Since the rate is given in inches per minute, we must convert the length from feet to inches to align the units. Knowing that

1\text{ foot} = 12\text{ inches}

, we multiply

6

by

12

.

2

Calculate the time in minutes to cut

72\text{ inches}

at a rate of

18\text{ inches/minute}

.

4\text{ minutes}

Dividing the total distance (

72\text{ inches}

) by the cutting rate (

18\text{ inches/minute}

) yields the time in minutes.

3

Convert the time from minutes to seconds.

240\text{ seconds}

Since

1\text{ minute} = 60\text{ seconds}

, multiplying

4

minutes by

60

gives the final time in seconds.

Key Concept

Performing multi-step unit conversions using rate and dimensional analysis.

Question 122Question →

A fiber-optic internet connection downloads data at a constant rate of $40$ megabits per second. There are $8$ megabits in $1$ megabyte and $1,000$ megabytes in $1$ gigabyte. At this rate, how many gigabytes of data can the connection download in $1$ hour?

Show answer & explanation

Answer: 18

Answer

The connection can download 18 gigabytes of data in 1 hour.

First, convert the download rate from megabits per second to megabytes per second:

40 \div 8 = 5

megabytes per second. Next, find the number of seconds in

1

hour:

60 \text{ minutes} \times 60 \text{ seconds} = 3,600

seconds. Multiply the rate in megabytes per second by the number of seconds in an hour to find the total megabytes downloaded:

5 \times 3,600 = 18,000

megabytes. Finally, convert megabytes to gigabytes:

18,000 \div 1,000 = 18

gigabytes.

Step-by-Step Solution

1

Convert the download speed from megabits per second to megabytes per second.

5

megabytes per second

Since

8\text{ megabits} = 1\text{ megabyte}

, divide

40

megabits per second by

8

.

2

Calculate the total number of seconds in

1

hour.

3,600

seconds

Since there are

60

minutes in

1

hour and

60

seconds in

1

minute, multiply

60

by

60

.

3

Determine the total megabytes downloaded in

1

hour.

18,000

megabytes

Multiply the rate of

5

megabytes per second by the elapsed time of

3,600

seconds.

4

Convert the total megabytes to gigabytes.

18

gigabytes

Since

1,000\text{ megabytes} = 1\text{ gigabyte}

, divide

18,000

megabytes by

1,000

.

Key Concept

Unit Conversions

Question 123Question →

A researcher records the daily high temperatures, in degrees Fahrenheit, for 9 consecutive days in a certain city: $72$ , $74$ , $75$ , $77$ , $78$ , $79$ , $81$ , $82$ , and $83$ . On the 10th day, the temperature was $x$ degrees Fahrenheit. If the median of the 10-day dataset is $78.5$ degrees Fahrenheit, which of the following could be the value of $x$ ?

75

76

78

80

Show answer & explanation

Answer: 80

Answer

80

To find the median of a dataset with 10 values, we sort the values in ascending order and average the 5th and 6th values. The original 9 values in sorted order are

72, 74, 75, 77, 78, 79, 81, 82, 83

. For the median of the 10-value dataset to be

78.5

, the 5th value must be

78

and the 6th value must be

79

. If the new value

x

is greater than or equal to

79

, the first 5 values will remain

72, 74, 75, 77, 78

, and the 6th value will be

79

. This results in a median of

\frac{78 + 79}{2} = 78.5

. Since

80

is greater than or equal to

79

, it could be the value of

x

.

Step-by-Step Solution

1

Determine the rule for the median of a dataset with 10 values.

The median of 10 values is the average of the 5th and 6th values when the dataset is sorted in ascending order.

This is the definition of the median for an even number of data points.

2

Set up an equation for the 5th and 6th values.

For the median to be

78.5

, the 5th and 6th values in the sorted list must average to

78.5

. Since all temperatures are integers, the 5th value must be

78

and the 6th value must be

79

.

The average of

78

and

79

is

\frac{78 + 79}{2} = 78.5

.

3

Find the inequality restriction on

x

.

The original sorted list has 5 values less than or equal to

78

:

72, 74, 75, 77, 78

. If the 10th value

x

is greater than or equal to

79

, it will be placed in position 6 or higher. The first 5 values will remain in positions 1 to 5, meaning the 5th value is

78

and the 6th value is

79

. Thus,

x

must be greater than or equal to

79

.

If

x

were less than

79

, it would shift

78

to the 6th position, resulting in a median that is less than or equal to

78

.

4

Identify the option that satisfies the restriction.

Among the options, only

80

is greater than or equal to

79

.

A value of

80

for

x

ensures the sorted list is

72, 74, 75, 77, 78, 79, 80, 81, 82, 83

, which has a median of

78.5

.

Key Concept

Calculating the median of a dataset with an even number of elements and understanding how the addition of a new element affects the positions of sorted values.

Question 124Question →

A commercial printing press uses ink at a constant rate of $0.24$ liters per minute. The density of the ink is $1.2$ grams per milliliter. Given that $1\text{ liter} = 1,000\text{ milliliters}$ and $1\text{ kilogram} = 1,000\text{ grams}$ , how many kilograms of ink does the press use in $8$ hours of continuous operation?

96.00

115.20

138.24

2.30

Show answer & explanation

Answer: $138.24$

Answer

The press uses

138.24

kilograms of ink in

8

hours of continuous operation.

To find the mass in kilograms, first convert the

8

hours of operation to minutes:

8 \times 60 = 480\text{ minutes}

. Next, find the volume of ink used in liters:

480 \times 0.24 = 115.2\text{ liters}

. Convert this volume to milliliters:

115.2 \times 1,000 = 115,200\text{ milliliters}

. Using the density of

1.2\text{ grams per milliliter}

, calculate the mass in grams:

115,200 \times 1.2 = 138,240\text{ grams}

. Finally, convert the mass to kilograms:

138,240 \div 1,000 = 138.24\text{ kilograms}

.

Step-by-Step Solution

1

Convert the operation time from hours to minutes.

8\text{ hours} \times 60\text{ minutes/hour} = 480\text{ minutes}

Since the ink consumption rate is given in liters per minute, the time must be in minutes to calculate the total volume of ink used.

2

Calculate the total volume of ink used in liters.

480\text{ minutes} \times 0.24\text{ liters/minute} = 115.2\text{ liters}

Multiply the total operation time by the rate of ink usage to find the total liters of ink consumed.

3

Convert the volume from liters to milliliters.

115.2\text{ liters} \times 1,000\text{ milliliters/liter} = 115,200\text{ milliliters}

Since the ink density is given in grams per milliliter, the volume must be in milliliters to find the mass.

4

Calculate the mass of the ink in grams using the density.

115,200\text{ milliliters} \times 1.2\text{ grams/milliliter} = 138,240\text{ grams}

Multiply the volume in milliliters by the density to find the mass in grams.

5

Convert the mass from grams to kilograms.

138,240\text{ grams} \div 1,000\text{ grams/kilogram} = 138.24\text{ kilograms}

Divide the mass in grams by

1,000

to convert it to kilograms.

Key Concept

Multi-step unit conversion involving rates, volume, density, and mass.

Question 125Question →

A shipping company records the weights, in pounds, of $10$ packages in a delivery batch. The table below shows the distribution of the weights.

Weight (pounds)	Frequency
$4$	$3$
$6$	$3$
$8$	$3$
$26$	$1$

If the heaviest package is removed from the batch, by how much will the mean weight of the remaining packages decrease, in pounds?

Show answer & explanation

Answer: 2

Answer

The mean weight of the remaining packages will decrease by

2

pounds.

The original mean weight of the

10

packages is calculated by finding the total weight and dividing it by

10

. The total weight is

(3 \times 4) + (3 \times 6) + (3 \times 8) + (1 \times 26) = 12 + 18 + 24 + 26 = 80

pounds, so the initial mean is

\frac{80}{10} = 8

pounds. After removing the heaviest package of

26

pounds, the remaining

9

packages have a total weight of

80 - 26 = 54

pounds. The new mean weight is

\frac{54}{9} = 6

pounds. The decrease in the mean weight is

8 - 6 = 2

pounds.

Step-by-Step Solution

1

Calculate the total weight of the

10

packages.

80

pounds

By summing the products of each weight and its corresponding frequency.

2

Calculate the initial mean weight.

8

pounds

By dividing the total weight of

80

pounds by the total count of

10

packages.

3

Calculate the sum of the weights of the remaining packages after removing the heaviest package.

54

pounds

By subtracting the weight of the heaviest package (

26

pounds) from the original total weight (

80

pounds).

4

Calculate the new mean weight of the remaining

9

packages.

6

pounds

By dividing the remaining weight (

54

pounds) by the new count of packages (

9

).

5

Find the decrease in the mean weight.

2

pounds

By subtracting the new mean weight (

6

pounds) from the original mean weight (

8

pounds).

Key Concept

Effect of outlier removal on the mean of a data distribution

Question 126Question →

An electric scooter travels at a constant speed of $15$ miles per hour. Given that $1\text{ mile} = 1.6$ kilometers, how many seconds does it take the scooter to travel $100$ meters?

Show answer & explanation

Answer: 15

Answer

The correct answer is 15. The scooter takes 15 seconds to travel 100 meters.

To find the time in seconds to travel 100 meters, we first convert the speed of 15 miles per hour to kilometers per hour by multiplying by 1.6, yielding 24 kilometers per hour. We then convert this speed to meters per second by multiplying by 1,000 (to convert kilometers to meters) and dividing by 3,600 (to convert hours to seconds), which simplifies to 20/3 meters per second. Finally, dividing the distance of 100 meters by this speed of 20/3 meters per second gives 15 seconds.

Step-by-Step Solution

1

Convert the speed of the scooter from miles per hour to kilometers per hour.

The speed is

24

kilometers per hour.

Since

1\text{ mile} = 1.6

kilometers, we multiply

15

by

1.6

to find the speed in kilometers per hour.

2

Convert the speed from kilometers per hour to meters per second.

The speed is

\frac{20}{3}

meters per second.

There are

1,000

meters in

1

kilometer and

3,600

seconds in

1

hour. Thus, the speed is

\frac{24 \times 1,000}{3,600} = \frac{20}{3}

meters per second.

3

Calculate the time required to travel

100

meters.

The time is

15

seconds.

Using the formula

\text{time} = \frac{\text{distance}}{\text{speed}}

, we divide

100

meters by the speed of

\frac{20}{3}

meters per second:

100 \div \frac{20}{3} = 15

seconds.

Key Concept

Converting compound units and applying rate formulas to solve for time.

Question 127Question →

The table below shows the insect populations in two different greenhouses, Greenhouse P and Greenhouse Q, monitored over a period of several weeks. The population in Greenhouse P grows linearly, while the population in Greenhouse Q grows exponentially.

Time (weeks)	Greenhouse P population	Greenhouse Q population
0	500	500
1	600	600
2	700	720

Based on the table, what is the positive difference between the population in Greenhouse Q and the population in Greenhouse P at week 3?

20

40

64

164

Show answer & explanation

Answer: 64

Answer

64

At week 3, the population of Greenhouse P (which grows by a constant 100 insects each week) is

500 + 100(3) = 800

. The population of Greenhouse Q (which grows by a constant factor of 1.2 each week) is

500(1.2)^3 = 864

. The positive difference between the two populations is

864 - 800 = 64

.

Step-by-Step Solution

1

Determine the linear growth model for the population in Greenhouse P.

The population in Greenhouse P starts at 500 at week 0 and increases to 600 at week 1, and 700 at week 2. This represents a constant increase of 100 insects per week. Therefore, the population of Greenhouse P at week 3 is

700 + 100 = 800

(or

500 + 100(3) = 800

).

To find the population of Greenhouse P at week 3, we must determine and apply its constant rate of change.

2

Determine the exponential growth model for the population in Greenhouse Q.

The population in Greenhouse Q starts at 500 at week 0 and increases to 600 at week 1, and 720 at week 2. The ratio between consecutive weeks is

\frac{600}{500} = 1.2

and

\frac{720}{600} = 1.2

. This represents a constant growth factor of 1.2 per week. Therefore, the population of Greenhouse Q at week 3 is

720 \times 1.2 = 864

(or

500 \times (1.2)^3 = 864

).

To find the population of Greenhouse Q at week 3, we must determine and apply its constant growth factor.

3

Calculate the positive difference between the two populations at week 3.

The positive difference is

864 - 800 = 64

.

The question asks for the positive difference between the population of Greenhouse Q and the population of Greenhouse P at week 3.

Key Concept

Linear and Exponential Growth

Question 128Question →

A botanist compares the heights, in centimeters, of two different varieties of sunflower, Variety A and Variety B, grown under identical greenhouse conditions. The table below shows the distribution of the heights for $6$ sunflowers of each variety.

Height (cm)	Variety A frequency	Variety B frequency
$110$	$0$	$1$
$115$	$0$	$1$
$120$	$2$	$1$
$121$	$2$	$0$
$122$	$2$	$1$
$127$	$0$	$1$
$132$	$0$	$1$

Which of the following statements correctly compares the standard deviations of the heights of Variety A and Variety B?

The standard deviation of Variety A is greater than the standard deviation of Variety B.

The standard deviation of Variety B is greater than the standard deviation of Variety A.

The standard deviations of the heights of Variety A and Variety B are equal.

The standard deviation of Variety B is equal to the mean height of Variety B.

Show answer & explanation

Answer: The standard deviation of Variety B is greater than the standard deviation of Variety A.

Answer

The standard deviation of Variety B is greater than the standard deviation of Variety A.

The correct statement is that the standard deviation of Variety B is greater than that of Variety A. The standard deviation measures how spread out the values in a dataset are from the mean. For Variety A, all data points are very close to the mean height of

121\text{ cm}

. For Variety B, the heights are spread much further from the mean, ranging from

110\text{ cm}

to

132\text{ cm}

. Since Variety B's data points show greater variability and are more spread out, the standard deviation of Variety B must be greater than the standard deviation of Variety A.

Step-by-Step Solution

1

Understand the concept of standard deviation.

Standard deviation measures the variability or spread of data points around their mean.

This sets up the conceptual framework for comparing the two varieties without needing complex calculations.

2

Analyze the distribution and spread of Variety A.

The heights of Variety A are clustered closely, with all values (

120

,

121

, and

122

) within

1\text{ cm}

of the mean height of

121\text{ cm}

.

This indicates that Variety A has a relatively low standard deviation due to its high concentration of data around the mean.

3

Analyze the distribution and spread of Variety B.

The heights of Variety B are spread out, ranging from

110\text{ cm}

to

132\text{ cm}

.

This demonstrates that Variety B has much higher variability around the mean than Variety A.

4

Compare the standard deviations based on the analyzed spreads.

Since Variety B is more spread out than Variety A, the standard deviation of Variety B must be greater than the standard deviation of Variety A.

This leads directly to the correct statement.

Key Concept

Standard deviation is a measure of how spread out the values in a dataset are from the mean.

Question 129Question →

A coffee shop owner records the number of customers served during each hour of an 8-hour shift. The recorded numbers of customers are $11$ , $13$ , $14$ , $15$ , $17$ , $18$ , $20$ , and $x$ . If the mean number of customers served per hour is equal to the median number of customers served per hour for this shift, and $x > 18$ , what is the value of $x$ ?

Show answer & explanation

Answer: 20

Answer

20

The correct value is 20. When

x > 18

, the 4th and 5th values of the sorted dataset are 15 and 17, giving a median of 16. Setting the mean, which is (108 + x)/8, equal to 16 yields x = 20, which is consistent with the condition

x > 18

.

Step-by-Step Solution

1

Determine the median of the dataset given the constraint

x > 18

.

The median of the dataset is

16

.

When the 8 values are sorted in ascending order, the first six values must be 11, 13, 14, 15, 17, and 18. The remaining two values, 20 and

x

, will occupy the 7th and 8th positions (in either order). Since there are 8 values, the median is the average of the 4th and 5th values:

\frac{15 + 17}{2} = 16

.

2

Set up the equation for the mean of the dataset.

The mean is represented by the expression

\frac{108 + x}{8}

.

The mean is the sum of all 8 values divided by 8:

\frac{11 + 13 + 14 + 15 + 17 + 18 + 20 + x}{8} = \frac{108 + x}{8}

.

3

Equate the mean to the median and solve for

x

.

x = 20

Setting the mean equal to the median gives

\frac{108 + x}{8} = 16

. Multiplying by 8 yields

108 + x = 128

. Subtracting 108 from both sides results in

x = 20

, which satisfies the condition

x > 18

.

Key Concept

Analyzing the mean and median of a dataset containing a variable constraint

Question 130Question →

A researcher wants to investigate the relationship between regular exercise and resting heart rate for adults residing in a certain city. The researcher obtains a list of all registered members of a large local fitness center and randomly selects 250 of these members to participate in a study. The study finds a strong negative correlation: members who reported exercising more than 5 hours per week had, on average, a significantly lower resting heart rate than those who reported exercising less than 5 hours per week. Which of the following is the most appropriate conclusion to draw from this study?

Exercising more than 5 hours per week causes a lower resting heart rate for all adults residing in the city.

Exercising more than 5 hours per week causes a lower resting heart rate for the members of this fitness center.

There is an association between exercising more than 5 hours per week and having a lower resting heart rate for the members of this fitness center, but a causal relationship cannot be established.

There is an association between exercising more than 5 hours per week and having a lower resting heart rate for all adults residing in the city, but a causal relationship cannot be established.

Show answer & explanation

Answer: There is an association between exercising more than 5 hours per week and having a lower resting heart rate for the members of this fitness center, but a causal relationship cannot be established.

Answer

There is an association between exercising more than 5 hours per week and having a lower resting heart rate for the members of this fitness center, but a causal relationship cannot be established.

The correct answer is the statement concluding that there is an association for the members of this fitness center, but a causal relationship cannot be established. Since the sample was randomly selected from the registered members of the local fitness center, the findings can be generalized to that population. However, because this is an observational study (no random assignment to exercise groups), a causal relationship cannot be established.

Step-by-Step Solution

1

Determine the target population that the sample represents.

The sample of 250 members was randomly selected from the registered members of a specific local fitness center.

Results of a statistical study can only be generalized to the population from which the sample was randomly selected. Because fitness center members are not representative of all adults in the city, the findings cannot be generalized to the entire city.

2

Evaluate the nature of the study design to check if causal conclusions can be made.

The study is observational because the researcher did not randomly assign participants to different exercise levels; the participants merely reported their existing habits.

A cause-and-effect relationship can only be established through a randomized controlled experiment. Observational studies can show association (correlation) but cannot establish causation.

3

Combine the limitations of generalization and causation to select the correct conclusion.

The study can only conclude that there is an association between exercise and resting heart rate for the members of this fitness center, with no causal claims allowed.

This is the only conclusion that respects both the sampling method and the observational design of the study.

Key Concept

Generalization and Causal Inference in Statistical Studies

Question 131Question →

An environmental study monitors the populations of two fish species in a lake. At the start of the study ( $t = 0$ ), Species A has a population of $1,200$ and its population decreases by $40$ each year. Species B has a population of $800$ and its population increases by $5\%$ each year. To the nearest whole number, what is the positive difference in the populations of the two species at year $t = 3$ ?

Show answer & explanation

Answer: 154

Answer

The positive difference in the populations of the two species at year

t = 3

is 154.

At

t = 3

, the population of Species A, which decreases linearly by 40 each year, is

1,200 - 40(3) = 1,080

. The population of Species B, which grows exponentially at

5\%

each year, is

800(1.05)^3 = 926.1

. The positive difference between these populations is

1,080 - 926.1 = 153.9

, which rounds to 154.

Step-by-Step Solution

1

Calculate the population of Species A at

t = 3

using a linear decay model.

A(3) = 1,080

Since Species A decreases by a constant number of 40 fish each year, its population follows the linear model

A(t) = 1,200 - 40t

. Substituting

t = 3

gives

1,200 - 120 = 1,080

.

2

Calculate the population of Species B at

t = 3

using an exponential growth model.

B(3) = 926.1

Since Species B increases by a constant percent of

5\%

each year, its population follows the exponential model

B(t) = 800(1.05)^t

. Substituting

t = 3

gives

800(1.05)^3 = 800(1.157625) = 926.1

.

3

Find the positive difference between the two populations and round to the nearest whole number.

154

The positive difference is

|A(3) - B(3)| = |1,080 - 926.1| = 153.9

. Rounding to the nearest whole number yields 154.

Key Concept

Distinguishing between linear models (constant rate of change per unit time) and exponential models (constant percent rate of change per unit time) to make projections.

Question 132Question →

The table below shows the distribution of the number of goals scored by a soccer team in each of its $15$ games last season.

Goals Scored	Number of Games
$0$	$3$
$1$	$5$
$2$	$4$
$3$	$2$
$9$	$1$

Which of the following statements best describes how the mean and the median of the number of goals scored will change if the game with $9$ goals is removed from the dataset?

Both the mean and the median values will remain unchanged.

The median value will be reduced, while the mean value will not change.

The mean value will be reduced, while the median value will not change.

Both the mean and the median values will be reduced.

Show answer & explanation

Answer: The mean value will be reduced, while the median value will not change.

Answer

The mean value will be reduced, while the median value will not change.

To find the effect of removing the game with

9

goals, we compare the mean and median before and after its removal. The initial dataset has

15

games. The median is the

8\text{th}

value, which is

1

. The initial mean is

(0 \times 3 + 1 \times 5 + 2 \times 4 + 3 \times 2 + 9 \times 1) / 15 = 28/15 \approx 1.87

. If the game with

9

goals is removed, there are

14

games remaining. The new median is the average of the

7\text{th}

and

8\text{th}

values, which are both

1

, so the median remains

1

. The new mean is

(0 \times 3 + 1 \times 5 + 2 \times 4 + 3 \times 2) / 14 = 19/14 \approx 1.36

. The mean value decreases (is reduced) while the median value remains the same.

Step-by-Step Solution

1

Calculate the initial mean and median of the dataset of

15

games.

The initial mean is approximately

1.87

and the initial median is

1

.

To establish the baseline values before removing any data.

2

Calculate the new mean and median of the dataset of

14

games after removing the game with

9

goals.

The new mean is approximately

1.36

and the new median is

1

.

To determine the values after the outlier is removed.

3

Compare the initial and new values for both the mean and median.

The mean decreases from approximately

1.87

to

1.36

, while the median remains

1

.

To select the statement that correctly describes the change in both statistical measures.

Key Concept

Effect of outliers on mean and median

Estimated Time:1m 30s

Question 133Question →

The daily electricity consumption, in kilowatt-hours (kWh), of a household was recorded for 7 consecutive days: $12$ , $12$ , $13$ , $14$ , $14$ , $15$ , and $45$ . If the household's electricity consumption of $45\text{ kWh}$ on the seventh day is removed from the dataset, which of the following statements best describes the effect on the mean and the median of the dataset?

The mean decreases by a greater amount than the median.

The median decreases by a greater amount than the mean.

The mean decreases, but the median increases.

The mean increases, but the median decreases.

Show answer & explanation

Answer: The mean decreases by a greater amount than the median.

Answer

The mean decreases by a greater amount than the median.

The correct answer is correct because calculating the original mean yields

\approx 17.86\text{ kWh}

and the original median yields

14\text{ kWh}

. After removing the outlier (

45\text{ kWh}

), the mean decreases to

\approx 13.33\text{ kWh}

(a change of

4.53\text{ kWh}

) and the median decreases to

13.5\text{ kWh}

(a change of

0.5\text{ kWh}

). Comparing these changes shows the mean decreased by a larger amount.

Step-by-Step Solution

1

Calculate the mean and median of the original 7-day dataset.

The original mean is

\frac{12 + 12 + 13 + 14 + 14 + 15 + 45}{7} = \frac{125}{7} \approx 17.86\text{ kWh}

. The sorted dataset is

12, 12, 13, 14, 14, 15, 45

, so the median is the middle value, which is

14\text{ kWh}

.

This establishes baseline values for the two measures.

2

Calculate the mean and median after removing the outlier of

45\text{ kWh}

.

The new mean is

\frac{12 + 12 + 13 + 14 + 14 + 15}{6} = \frac{80}{6} \approx 13.33\text{ kWh}

. The new sorted dataset is

12, 12, 13, 14, 14, 15

, so the median is the average of the two middle values:

\frac{13 + 14}{2} = 13.5\text{ kWh}

.

This provides the new values for comparison.

3

Compare the changes in the mean and the median.

The mean decreased by

17.86 - 13.33 = 4.53\text{ kWh}

. The median decreased by

14 - 13.5 = 0.5\text{ kWh}

. The decrease in the mean is greater.

This confirms the statement that describes the relative changes.

Key Concept

Effect of outliers on mean and median

Question 134Question →

A researcher wants to estimate the total number of households in a city of $40,000$ households that own at least one hybrid vehicle. The researcher randomly selects a sample of households to contact for a survey. The survey has a response rate of $40\%$ . Of the households that responded, $15\%$ reported owning at least one hybrid vehicle. A follow-up study of a random sample of the non-respondents indicates that the proportion of non-respondents who own at least one hybrid vehicle is half the proportion of the respondents who do. Based on the results of the survey and the follow-up study, what is the estimated total number of households in the city that own at least one hybrid vehicle?

Show answer & explanation

Answer: 4200

Answer

The estimated total number of households in the city that own at least one hybrid vehicle is 4,200.

The correct answer of 4,200 is found by dividing the population of 40,000 households into two strata: respondents (40%, or 16,000 households) and non-respondents (60%, or 24,000 households). The number of hybrid-owning households among respondents is estimated as 15% of 16,000, which is 2,400. The proportion of hybrid owners among non-respondents is half of the respondents' rate, or 7.5%. The number of hybrid-owning households among non-respondents is estimated as 7.5% of 24,000, which is 1,800. Summing these estimates yields 2,400 + 1,800 = 4,200.

Step-by-Step Solution

1

Calculate the size of the respondent and non-respondent strata in the population.

Respondent stratum:

16,000

households; Non-respondent stratum:

24,000

households.

Since the survey's response rate is

40\%

, we estimate that

40\%

of the

40,000

households are potential respondents, and the remaining

60\%

(

24,000

households) are potential non-respondents.

2

Determine the proportion of hybrid vehicle owners in each stratum.

Respondent proportion:

15\%

(or

0.15

); Non-respondent proportion:

7.5\%

(or

0.075

).

The respondent proportion is given as

15\%

. The follow-up study indicates the non-respondent proportion is half of the respondent proportion, which is

\frac{1}{2} \times 15\% = 7.5\%

.

3

Calculate the estimated number of households owning a hybrid vehicle in each stratum and sum them.

2,400 + 1,800 = 4,200

households.

Multiply each stratum size by its respective proportion:

16,000 \times 0.15 = 2,400

and

24,000 \times 0.075 = 1,800

. Adding these gives the total estimate of

4,200

.

Key Concept

Generalizing from stratified random samples to estimate population parameters while accounting for non-response bias.

Question 135Question →

An agricultural scientist is tracking the biomass, in grams, of two different plant species under controlled conditions. Species A starts with a biomass of $250$ grams and increases by $12$ grams each day. Species B starts with a biomass of $180$ grams and its biomass increases exponentially by a constant daily percentage. On day $5$ , the biomass of Species A and Species B is equal. What is the daily percentage increase in the biomass of Species B, to the nearest tenth of a percent?

11.5\%

14.4\%

34.4\%

10.0\%

Show answer & explanation

Answer: $11.5\%$

Answer

The correct answer is

11.5\%

.

The correct answer is

11.5\%

. First, find the biomass of Species A on day

5

using the linear equation

250 + 12(5) = 310

grams. Because the biomass of both species is equal on day

5

, set up the exponential equation for Species B as

180(1 + r)^5 = 310

, where

r

is the daily rate of increase. Divide both sides by

180

to get

(1 + r)^5 \approx 1.7222

. Taking the fifth root of both sides yields

1 + r \approx 1.1148

, which means

r \approx 0.1148

, or

11.5\%

.

Step-by-Step Solution

1

Calculate the biomass of Species A on day

5

.

The biomass of Species A on day

5

is

250 + 12(5) = 310

grams.

This establishes the common biomass value that both species reach on day

5

using the linear growth model.

2

Set up the exponential growth equation for Species B.

The equation is

180(1 + r)^5 = 310

, where

r

is the daily growth rate.

Species B grows exponentially, starting at

180

grams and reaching

310

grams on day

5

.

3

Solve for the daily growth rate,

r

.

(1 + r)^5 = \frac{310}{180} \approx 1.7222

, which yields

1 + r \approx (1.7222)^{1/5} \approx 1.1148

, so

r \approx 0.1148

or

11.5\%

.

Isolating

r

by taking the fifth root of both sides gives the daily percentage increase.

Key Concept

Linear vs. Exponential Growth Models

Question 136Question →

An online streaming service tracks the number of subscribers in two regions, Region X and Region Y. At the beginning of 2020 ( $t = 0$ ), Region X had $12,000$ subscribers, and the number of subscribers in Region X increased by a constant amount of $2,500$ each year. Region Y had $8,000$ subscribers at the beginning of 2020, and the number of subscribers in Region Y increased by a constant $15\%$ each year. What is the positive difference between the number of subscribers in Region X and the number of subscribers in Region Y at the beginning of 2022 ( $t = 2$ )?

Show answer & explanation

Answer: 6420

Answer

The positive difference between the subscriber counts in Region X and Region Y at the beginning of 2022 is

6,420

.

To find the positive difference, we first calculate the subscriber counts for both regions at

t = 2

. Region X grows linearly at a constant rate of

2,500

subscribers per year, so its population at

t = 2

is

12,000 + 2,500(2) = 17,000

. Region Y grows exponentially at a constant rate of

15\%

per year, so its population at

t = 2

is

8,000 \times (1.15)^2 = 10,580

. The positive difference between the subscriber counts is

17,000 - 10,580 = 6,420

.

Step-by-Step Solution

1

Calculate the subscriber count for Region X at

t = 2

.

17,000

Since Region X grows linearly by a constant amount of

2,500

subscribers per year, the number of subscribers at

t = 2

is given by

12,000 + 2,500(2) = 17,000

.

2

Calculate the subscriber count for Region Y at

t = 2

.

10,580

Since Region Y grows exponentially by a constant rate of

15\%

per year, the number of subscribers at

t = 2

is given by

8,000 \times (1.15)^2 = 10,580

.

3

Calculate the positive difference between the two subscriber counts.

6,420

Subtract the smaller subscriber count from the larger subscriber count:

17,000 - 10,580 = 6,420

.

Key Concept

Distinguishing between linear growth (constant rate of change per unit of time) and exponential growth (constant percent rate of change per unit of time) to solve real-world problems.

Question 137Question →

A county health department is evaluating a new public health program designed to lower blood pressure among residents with hypertension. The county has $120,000$ adult residents, of whom $30,000$ have been diagnosed with hypertension. The department selects a random sample of $400$ adults diagnosed with hypertension from a list of patients at a major county clinic to participate in the program. After six months, $280$ of the participants show a significant reduction in blood pressure. Which of the following is the most appropriate conclusion?

The program is likely to be associated with a significant reduction in blood pressure among adult residents with hypertension who are patients at this major county clinic.

The program is likely to be associated with a significant reduction in blood pressure for all adult residents of the county.

The program is likely to be associated with a significant reduction in blood pressure for all

30,000

adult residents of the county who have hypertension.

It can be estimated that exactly

2,100

adult residents in the county with hypertension would show a significant reduction in blood pressure if they participated in the program.

Show answer & explanation

Answer: The program is likely to be associated with a significant reduction in blood pressure among adult residents with hypertension who are patients at this major county clinic.

Answer

The program is likely to be associated with a significant reduction in blood pressure among adult residents with hypertension who are patients at this major county clinic.

The correct answer is correct because the sample of participants was randomly selected from patients diagnosed with hypertension at a specific major county clinic. Therefore, the findings can be generalized to the population of patients with hypertension at this specific clinic.

Step-by-Step Solution

1

Identify the population from which the sample was selected.

The sample of

400

adults was randomly selected from the patients diagnosed with hypertension at a single major county clinic.

Results of a study can only be generalized to the population from which the sample was randomly selected.

2

Determine the limits of generalization for the study's findings.

Because the sample was only drawn from patients at this specific clinic, the findings cannot be generalized to all adults in the county or even to all adults in the county with hypertension.

Patients at one clinic may not be representative of the entire county's population or all individuals with hypertension in the county.

3

Evaluate the options based on these generalization rules and mathematical accuracy.

The option proposing a significant reduction in blood pressure for patients at this clinic is valid. The options generalizing to the entire county or all hypertensive county residents are invalid due to sampling bias. Stating that exactly

2,100

residents would benefit is both an invalid generalization and mathematically incorrect since

70\%

of

30,000

is

21,000

, not

2,100

.

This confirms the correct option matches the sampling frame and avoids unwarranted extrapolation and arithmetic errors.

Key Concept

Generalization in observational studies and surveys requires that the sample be randomly selected from the target population.

Question 138Question →

A set of 5 positive integers has a mean of 12, a median of 10, and a unique mode of 8. What is the maximum possible value of the largest integer in this set?

Show answer & explanation

Answer: 23

Answer

The maximum possible value of the largest integer in the set is 23.

To find the maximum possible value of the largest integer in a set of 5 positive integers with a mean of 12, a median of 10, and a unique mode of 8: First, calculate the total sum of the integers, which is

12 \times 5 = 60

. Let the sorted integers be

x_1 \le x_2 \le x_3 \le x_4 \le x_5

. The median is the middle term, so

x_3 = 10

. Since 8 is the unique mode and is less than the median, it must appear at least twice in the first two slots, so

x_1 = 8

and

x_2 = 8

. The sum of the remaining two integers is

60 - (8 + 8 + 10) = 34

. To maximize the largest integer

x_5

, we must minimize

x_4

. Since the integers are sorted,

x_4 \ge x_3 = 10

. However, if

x_4 = 10

, then 10 would appear twice, making it a second mode alongside 8, which violates the unique mode condition. Thus, the smallest possible integer value for

x_4

is 11, which gives a maximum possible value of

34 - 11 = 23

for the largest integer.

Step-by-Step Solution

1

Calculate the sum of the five integers.

The sum of the five integers is 60.

Since the mean of 5 numbers is 12, their sum is

12 \times 5 = 60

.

2

Set up the ordered list of integers and identify the median.

For integers

x_1 \le x_2 \le x_3 \le x_4 \le x_5

, the median is

x_3 = 10

.

The median of 5 numbers in ordered sequence is the third number.

3

Determine the values of the first two integers using the mode constraint.

x_1 = 8

and

x_2 = 8

.

The mode must be unique and equal to 8. Since 8 is less than the median 10, 8 must occupy the first two spots to appear more than once.

4

Write the sum equation for the remaining unknown integers.

x_4 + x_5 = 34

.

Since the total sum is 60, we subtract the known values:

60 - (8 + 8 + 10) = 34

.

5

Minimize the fourth integer to maximize the fifth integer while keeping the unique mode of 8.

The minimum value for

x_4

is 11, which gives the maximum value for

x_5

as 23.

We must have

x_4 \ge 10

due to ordering. If

x_4 = 10

, 10 would be a mode, so the minimum valid integer for

x_4

is 11.

Key Concept

Using measures of center (mean, median) and measures of frequency (mode) to determine constraints on individual data values in a data distribution.

Alternative Method

Instead of setting up inequalities, one can test integers starting from the maximum mathematical limit if there were no mode constraint (which is 24, since

x_4 \ge 10

, giving

60 - 26 - 10 = 24

). Testing 24 yields the set {8, 8, 10, 10, 24}, which has two modes. The next highest value to test is 23, which yields the valid set {8, 8, 10, 11, 23}.

Estimated Time:1m 30s

Question 139Question →

A sociologist is studying the print newspaper subscription habits of residents in a city. The table below shows the total number of households, the sample size, and the number of households in each sample that report subscribing to a print newspaper for the city's two districts.

District	Total households	Sample size	Subscribing households in sample
West	$10,000$	$200$	$35$
East	$14,000$	$250$	$60$

Based on the results of these samples, what is the estimated total number of households in the city that subscribe to a print newspaper?

Show answer & explanation

Answer: 5110

Answer

5110

To find the total estimated number of households in the city that subscribe to a print newspaper, we must estimate the subscriptions for each district separately and then sum them. For District West, the proportion of households subscribing is

\frac{35}{200} = 0.175

. Multiplying this by the total number of households in District West (

10,000

) gives an estimate of

1,750

subscribing households. For District East, the proportion of households subscribing is

\frac{60}{250} = 0.24

. Multiplying this by the total number of households in District East (

14,000

) gives an estimate of

3,360

subscribing households. Adding these two estimates together (

1,750 + 3,360

) results in a total estimate of

5,110

households.

Step-by-Step Solution

1

Calculate the estimated number of print newspaper subscriptions in District West.

1,750 households

The proportion of subscribing households in the District West sample is applied to the total number of households in District West.

2

Calculate the estimated number of print newspaper subscriptions in District East.

3,360 households

The proportion of subscribing households in the District East sample is applied to the total number of households in District East.

3

Find the total estimated number of subscribing households in the city.

5,110 households

Adding the estimates from both districts yields the overall population estimate.

Key Concept

Estimating population parameters from stratified random samples

Question 140Question →

A scientist is monitoring the population of two different bacteria cultures, Culture A and Culture B, in a laboratory. The table below shows the cell counts of the two cultures at $1$ hour and $3$ hours after the start of the experiment ( $t=0$ hours).

Time ( $t$ , hours)	Culture A Cell Count	Culture B Cell Count
1	1,200	1,200
3	2,700	2,700

It is known that the population of Culture A grows linearly, while the population of Culture B grows exponentially. If $A_0$ is the initial population of Culture A at $t=0$ hours, and $B_0$ is the initial population of Culture B at $t=0$ hours, what is the value of $B_0 - A_0$ ?

150

350

250

50

Show answer & explanation

Answer: 350

Answer

The correct difference between the initial population of Culture B and Culture A is 350.

To find the initial values, we must model both cultures. For the linear growth of Culture A, the population increases by a constant amount each hour. The change over two hours is

2,700 - 1,200 = 1,500

cells, which corresponds to a growth rate of

750

cells per hour. Subtracting this hourly increase from the population at

1

hour (

1,200

) gives an initial population

A_0 = 450

. For the exponential growth of Culture B, the population increases by a constant factor each hour. The ratio of the population at

3

hours to the population at

1

hour is

\frac{2,700}{1,200} = 2.25

, which represents the growth factor squared (

b^2

). Taking the square root gives a growth factor of

b = 1.5

per hour. Dividing the population at

1

hour (

1,200

) by this factor gives an initial population

B_0 = 800

. The value of

B_0 - A_0

is

800 - 450 = 350

.

Step-by-Step Solution

1

Find the constant rate of change and the initial value

A_0

of the linear model (Culture A).

The hourly rate of change is

m = \frac{2700 - 1200}{3 - 1} = 750

cells per hour. The initial population is

A_0 = 1200 - 750(1) = 450

cells.

Linear growth has a constant absolute increase per unit of time, modeled by

A(t) = A_0 + m \cdot t

.

2

Find the constant growth factor and the initial value

B_0

of the exponential model (Culture B).

The growth factor squared is

b^2 = \frac{2700}{1200} = 2.25 \implies b = 1.5

. The initial population is

B_0 = \frac{1200}{1.5^1} = 800

cells.

Exponential growth has a constant multiplier per unit of time, modeled by

B(t) = B_0 \cdot b^t

.

3

Calculate the difference between the two initial values

B_0 - A_0

.

B_0 - A_0 = 800 - 450 = 350

.

The question asks for the value of the difference between the initial population of Culture B and the initial population of Culture A.

Key Concept

Linear and Exponential Growth

Problem-Solving and Data Analysis

Answer

Step-by-Step Solution

Key Concept

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Common Mistakes

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Answer

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Answer

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Answer

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Answer

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Answer

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Answer

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Answer

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Answer

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Answer

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Answer

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Common Mistakes

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Common Mistakes

Answer

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