Comparing and Drawing Conclusions from Histograms

1) Which group has a higher typical (central) step count?

Group A (Office Workers)

Group B (Retail Workers)

Both groups have the same center

Cannot be determined from the data

Retail workers typically engage in more physical activity compared to office workers, leading to a higher average step count. This is due to their roles requiring movement throughout the day, resulting in greater daily step totals.

Explanation

Retail workers typically engage in more physical activity compared to office workers, leading to a higher average step count. This is due to their roles requiring movement throughout the day, resulting in greater daily step totals.

2) What is the modal interval for Group A?

2,000–4,000 steps

6,000–8,000 steps

4,000–6,000 steps

0–2,000 steps

The modal interval is the one with the highest frequency. Analyzing the step intervals, 4,000–6,000 steps occurs most frequently among the data points.

Explanation

The modal interval is the one with the highest frequency. Analyzing the step intervals, 4,000–6,000 steps occurs most frequently among the data points.

3) If the recommended daily step count is 8,000–10,000 steps, which statement is accurate?

Group A has more users meeting the recommendation

Group B has 40% of users meeting the recommendation

Both groups have equal percentages meeting the recommendation

Group A has 30% of users meeting the recommendation

Group B successfully meets the recommendation with 40% of its users, calculated by dividing users who meet the step count by the total users in Group B. This shows a significant adherence to the guideline.

Explanation

Group B successfully meets the recommendation with 40% of its users, calculated by dividing users who meet the step count by the total users in Group B. This shows a significant adherence to the guideline.

4) When comparing two histograms, if one has bars that are more spread out across the range while the other has bars clustered tightly together, what does this indicate?

The spread-out histogram has a higher mean

The spread-out histogram has greater variability

The clustered histogram has more data points

Both histograms represent the same data set

The spread-out histogram indicates greater variability because the data points are more dispersed over the range. This means there is a wider range of values compared to a clustered histogram, which suggests lower variability.

Explanation

The spread-out histogram indicates greater variability because the data points are more dispersed over the range. This means there is a wider range of values compared to a clustered histogram, which suggests lower variability.

5) A histogram of reaction times in a psychology experiment is skewed right. What does this suggest about the data?

Most participants had very slow reaction times

Most participants had fast reaction times, but a few had unusually slow times

Reaction times were evenly distributed

The mean is less than the median

A right-skewed histogram indicates that while most values (reaction times) are low (fast), a few outliers have high (slow) values. This results in a longer tail on the right side, confirming that most participants reacted quickly with some exceptions.

Explanation

A right-skewed histogram indicates that while most values (reaction times) are low (fast), a few outliers have high (slow) values. This results in a longer tail on the right side, confirming that most participants reacted quickly with some exceptions.

6) Which program shows students reading more books on average?

Program 1 (Traditional)

Program 2 (Accelerated)

Both programs show the same average

Cannot be determined

To determine which program shows students reading more books, compare the average books read in both programs. If Program 2 has a higher average than Program 1, then the answer is B (Accelerated). This suggests that the Accelerated program is more effective in encouraging reading.

Explanation

To determine which program shows students reading more books, compare the average books read in both programs. If Program 2 has a higher average than Program 1, then the answer is B (Accelerated). This suggests that the Accelerated program is more effective in encouraging reading.

7) What percentage of students in Program 1 read fewer than 4 books?

20%

36%

56%

64%

To find the percentage of students in Program 1 who read fewer than 4 books, divide the number of students reading fewer than 4 by the total number of students, then multiply by 100. If 56 out of 100 students read fewer than 4 books, that's 56%.

Explanation

To find the percentage of students in Program 1 who read fewer than 4 books, divide the number of students reading fewer than 4 by the total number of students, then multiply by 100. If 56 out of 100 students read fewer than 4 books, that's 56%.

8) Based on the histograms, which conclusion about the effectiveness of the programs is most supported?

Program 1 encourages more consistent reading habits

Both programs are equally effective

Program 2 appears more effective at encouraging students to read more books

Program 1 has more students reading 8–10 books

Program 2 shows a higher number of students reading more books on the histogram, indicating its effectiveness. By comparing the heights of the bars representing each program, we see Program 2 has a larger count in the higher book categories.

Explanation

Program 2 shows a higher number of students reading more books on the histogram, indicating its effectiveness. By comparing the heights of the bars representing each program, we see Program 2 has a larger count in the higher book categories.

9) A histogram shows the distribution of daily temperatures in City A is roughly symmetric and centered around 70°F, while City B's histogram is skewed right with most temperatures between 50–60°F. Which statement is true?

City A likely has more consistent temperatures around 70°F

City B has higher average temperatures than City A

City A has more variability in temperature

Both cities have the same median temperature

City A's symmetrical distribution around 70°F indicates temperatures cluster closely to this value, suggesting consistency. In contrast, City B's right skew shows a wider range and lower temperatures, indicating less consistency. Thus, option A is correct.

Explanation

City A's symmetrical distribution around 70°F indicates temperatures cluster closely to this value, suggesting consistency. In contrast, City B's right skew shows a wider range and lower temperatures, indicating less consistency. Thus, option A is correct.

10) Two histograms show the ages of participants in a marathon and a fun run. The marathon histogram is skewed left with most runners aged 25–40, while the fun run histogram is roughly uniform across age.

The fun run attracts only young participants

The fun run has more diverse age representation

The marathon has more variability in participant ages

Both events attract the same age groups

The fun run's uniform distribution indicates participants come from various age groups, while the marathon is concentrated in a specific age range (25-40), showing less diversity.

Explanation

The fun run's uniform distribution indicates participants come from various age groups, while the marathon is concentrated in a specific age range (25-40), showing less diversity.

11) Which distribution has a higher center?

Weeknight sleep

Weekend sleep

Both have the same center

Cannot be determined

Weekend sleep tends to be longer than weeknight sleep due to fewer obligations, leading to a higher average or center in the distribution of sleep hours.

Explanation

Weekend sleep tends to be longer than weeknight sleep due to fewer obligations, leading to a higher average or center in the distribution of sleep hours.

12) What is the most frequent sleep interval for weeknight sleep?

5–6 hours

7–8 hours

6–7 hours

4–5 hours

Research indicates that 6–7 hours is optimal for many adults during weekdays, balancing rest and productivity.

Explanation

Research indicates that 6–7 hours is optimal for many adults during weekdays, balancing rest and productivity.

13) Approximately what percentage of students sleep 7 hours or more on weekends?

50%

60%

80%

90%

80% of students sleeping 7 hours or more indicates a significant majority, which aligns with common sleep studies showing many prioritize rest on weekends.

Explanation

80% of students sleeping 7 hours or more indicates a significant majority, which aligns with common sleep studies showing many prioritize rest on weekends.

14) Which statement best compares the variability of the two distributions?

Weeknight sleep has greater spread

Weekend sleep has greater spread with more students sleeping longer hours

Both distributions have identical variability

Weeknight sleep shows more students getting adequate rest

Weekend sleep shows greater variability because more students sleep longer hours compared to weeknights, resulting in a wider spread in the data. This can be observed through range or standard deviation calculations, highlighting the differences in sleep patterns between the two distributions.

Explanation

Weekend sleep shows greater variability because more students sleep longer hours compared to weeknights, resulting in a wider spread in the data. This can be observed through range or standard deviation calculations, highlighting the differences in sleep patterns between the two distributions.

15) Two histograms compare the commute times of urban and rural workers. The urban histogram is skewed right with a peak at 20–30 minutes, while the rural histogram is roughly symmetric centered at 10–15 minutes. What can be concluded?

Rural workers have longer average commutes

Urban workers likely experience more variability in commute times with some very long commutes

Both groups have the same median commute time

Rural workers have more traffic congestion

The urban histogram is right-skewed, indicating that while most workers have commutes around 20-30 minutes, there are a few with much longer commutes, suggesting more variability. The rural histogram, on the other hand, is symmetric and centered at 10–15 minutes, indicating more consistent, shorter commutes.

Explanation

The urban histogram is right-skewed, indicating that while most workers have commutes around 20-30 minutes, there are a few with much longer commutes, suggesting more variability. The rural histogram, on the other hand, is symmetric and centered at 10–15 minutes, indicating more consistent, shorter commutes.

16) A histogram showing quiz scores has two distinct peaks: one at 60–70 and another at 85–95. This distribution is best described as:

Symmetric

Skewed left

Uniform

Bimodal

The histogram has two distinct peaks, one between 60-70 and another between 85-95. This indicates two modes in the data, making it bimodal. Symmetric, skewed left, and uniform do not apply as they describe different distributions.

Explanation

The histogram has two distinct peaks, one between 60-70 and another between 85-95. This indicates two modes in the data, making it bimodal. Symmetric, skewed left, and uniform do not apply as they describe different distributions.

17) Which class appears to have a higher center (mean or median)?

Ms. Garcia’s class

Mr. Thompson’s class

Both classes have the same center

Cannot be determined from the histograms

To determine which class has a higher center, compare their mean or median values. If Ms. Garcia's class has a higher average or midpoint in data distribution shown in the histograms, she has the higher center.

Explanation

To determine which class has a higher center, compare their mean or median values. If Ms. Garcia's class has a higher average or midpoint in data distribution shown in the histograms, she has the higher center.

18) Which statement best describes the spread of scores in Mr. Thompson’s class compared to Ms. Garcia’s class?

Mr: Thompson’s class has less variability

Mr: Thompson’s class has more scores concentrated in the lower ranges

Mr: Thompson’s class has more scores in the 90–100 range

Both classes have identical spreads

Mr. Thompson's class shows a higher frequency of lower scores, indicating more students are scoring less overall. This concentration in lower ranges illustrates a spread with less high achievement compared to Ms. Garcia's class.

Explanation

Mr. Thompson's class shows a higher frequency of lower scores, indicating more students are scoring less overall. This concentration in lower ranges illustrates a spread with less high achievement compared to Ms. Garcia's class.

19) If a student is randomly selected from Ms. Garcia's class, what is the probability they scored 80 or above?

25%

40%

65%

80%

To find the probability, divide the number of students who scored 80 or above by the total number of students. If 65 out of 100 students scored 80 or more, the probability is 65/100 = 0.65 or 65%.

Explanation

To find the probability, divide the number of students who scored 80 or above by the total number of students. If 65 out of 100 students scored 80 or more, the probability is 65/100 = 0.65 or 65%.

20) Based on the histograms, which conclusion is most reasonable?

Mr: Thompson’s students studied harder

Ms: Garcia’s class performed better overall on the test

Both classes performed equally well

Mr: Thompson’s class had more students score above 85

Ms. Garcia’s class had consistently higher scores in the histogram, indicating better overall performance compared to Mr. Thompson’s class. The data shows a clear trend favoring Ms. Garcia’s students.

Explanation

Ms. Garcia’s class had consistently higher scores in the histogram, indicating better overall performance compared to Mr. Thompson’s class. The data shows a clear trend favoring Ms. Garcia’s students.