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Use the following data to answer the next five exercises: Two researchers are gathering data on hours of video games played by school-aged children and young adults. They each randomly sample different groups of 150 students from the same school. They collect the following data. $$\begin{array}{|l|l|l|l|l|l|l|}\hline \text { Hours Played per week } & {\text { Frequency }} & {\text { Relative Frequency }} & {\text { Cumulative Relative Frequency }} \\ \hline 0-2 & {26} & {0.17} & {0.17} \\ \hline 2-4 & {30} & {0.20} & {0.37} \\ \hline 4-6 & {49} & {0.33} & {0.70} \\ \hline 6-8 & {25} & {0.17} & {0.87} \\ \hline 8-10 & {12} & {0.8} & {0.95} \\ \hline 10-12 & {8} & {0.05} & {1} \\ \hline\end{array}$$ Table 1.29 Researcher A $$\begin{array}{|l|l|l|l|}\hline \text { Hours Played per week } & {\text { Frequency }} & {\text { Relative Frequency }} & {\text { Cumulative Relative Frequency }} \\ \hline 0-2 & {0.48} & {0.32} & {0.32} \\ \hline 2-4 & {51} & {0.34} & {0.66} \\ \hline 4-6 & {24} & {0.16} & {0.82} \\ \hline 6-8 & {12} & {0.08} & {0.90} \\ \hline 8-10 & {11} & {0.07} & {0.97} \\ \hline 10-12 & {4} & {0.03} & {1} \\ \hline \end{array}$$ Table 1.30 Researcher B As part of a way to reward students for participating in the survey, the researchers gave each student a gift card to a video game store. Would this affect the data if students knew about the award before the study?

Step by step solution

01

Understand the Exercise

The question asks whether informing participants about a reward before collecting data could affect the results of a survey about hours spent playing video games. Both researchers collected data from separate groups and recorded frequencies and relative frequencies of video game usage.

02

Consider Effects of Incentives

When participants know about a reward beforehand, they might modify their behavior or responses to qualify for the reward, or they might feel more encouraged to participate, possibly impacting honest self-reporting.

03

Analyze Potential Biases

Participants who are aware of being rewarded might overstate or understate their gaming hours. This misreporting can introduce bias, leading to inaccurate representations of true gaming habits.

04

Compare Bias in Responses

Although the specific data isn't directly analyzed here, general research principles suggest that pre-knowledge of incentives can skew data in favor of participants either exaggerating or minimizing their responses to align with what they perceive might influence their reward.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Frequency Distribution

Frequency distribution is a helpful way to represent data visually. Imagine you are counting and sorting different types of candies into their categories. Similarly, frequency distribution sorts data into specific interval ranges, just like hours of video game play are sorted into 0-2 hours, 2-4 hours, and so on. Each category or interval has a count of how many occurrences fall into that range, which is called frequency.

• In the given example of two researchers, you can see frequencies like 26 students played 0-2 hours, and 30 students played 2-4 hours a week from one table.
• This gives a clear snapshot of how often each interval occurs, helping to make sense of large datasets by simplifying it into smaller, more manageable parts.

Learning these frequencies helps to quickly see which categories are most and least common, giving a focused perspective at a glance.

Relative Frequency

Relative frequency helps us understand proportions within the data by comparing frequencies to the total. It is like looking at a pie and figuring out how much of it represents chocolate chips if each chip is a frequency count. Relative frequency takes the actual count of occurrences in one category and divides it by the total number of observations.

• For instance, if 30 students were in the 2-4 hours category out of 150 total, the relative frequency would be computed as \( \frac{30}{150} = 0.20 \).
• This means 20% of students fall in this range, providing insight into the proportion of the total each category represents.

You can tell which category takes up the biggest slice of the pie (or data set) in relative terms compared to others, allowing better understanding of the distribution and making comparisons easier.

Cumulative Relative Frequency

Cumulative relative frequency is like climbing a staircase where each step you take adds up the ones you've already climbed. This concept helps us understand the running total of relative frequencies, which tells us not just about individual categories, but about the total proportion up to that category.

• For example, if the cumulative relative frequency of 2-4 hours is 0.37, it means 37% of students play video games for up to 4 hours a week.
• It gives a cumulative sense of all categories up to a certain point rather than isolated intervals alone.

This approach is particularly useful to see overall trends and patterns as you progress through the data, aiding in grasping the full scope and build-up of data relationships.