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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 Would the sample size be large enough if the population is school-aged children and young adults in the United States?

Step by step solution

01

Understand the Problem

We need to determine if the sample size is sufficiently large for two researchers who are gathering data on video game playtime among school-aged children and young adults in the United States. Each researcher has sampled 150 students.

02

Identify Sample Size and Population

The total population of school-aged children and young adults in the United States is a very large number, possibly in the millions. Each sample consists of 150 students, which was gathered by the researchers separately.

03

Assess Sample Size Relevance

To assess if a sample size is sufficient, it is often compared to the entire population. For large populations like that of school-aged children in the United States, a sample size of 150 is relatively small.

04

Consider Statistical Sampling Methods

Typically, larger sample sizes or a series of smaller samples are considered when aiming to represent a very large population like the United States. Samples should be random and representative to generalize the findings to the entire population.

05

Conclusion

For statistical significance and accuracy in representing a population as vast as school-aged children and young adults in the U.S., a sample larger than 150 may be necessary unless certain confidence levels and error margins are to be tolerated.

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Key Concepts

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

Representative Sampling

In the realm of statistical studies, representative sampling plays a crucial role in ensuring the accuracy of the results. It refers to the method used to select a subset of individuals from a larger population in a way that the sample reflects the diverse characteristics of the whole group.
Imagine wanting to know the average time spent on video games by all school-aged children in the U.S. It's practically impossible to ask every single one. Instead, researchers use representative samples to make educated guesses about the larger group.

• A representative sample captures the essence of the entire population, providing dependable data.
• Key aspects like age distribution, gender, and geographical location should mirror those of the broader population.
• The core goal is to minimize bias. Bias can distort findings if certain groups are over or underrepresented in the sample.

In conclusion, while our researchers each sampled 150 students, one must scrutinize whether this group accurately mirrors the broader population of school-aged children across the United States. This ensures that their findings can genuinely be applied to the entire population.

Relative Frequency

Relative frequency is a simple yet insightful concept in statistics. It shows how often a particular data point occurs, relative to the total number of observations. This gives researchers a way to understand the prevalence of various categories or outcomes.

• It is calculated by taking the frequency of a particular event divided by the total number of observations.
• For example, if you have a frequency of 30 students playing for 2-4 hours per week out of a sample of 150, the relative frequency is calculated as 30/150 = 0.20, or 20%.

The power of relative frequency is in its ability to transform raw data into something more meaningful. By comparing these proportions, we can easily see how different intervals stack up against each other.
Moreover, researchers often use relative frequency to provide context within their results, helping stakeholders understand the data without delving deeply into raw numbers.

Cumulative Frequency

Cumulative frequency allows us to comprehend data through an "accumulation" perspective. Instead of looking at isolated events, it gives us a growing tally of occurrences as we move through a dataset. This helps in identifying overall trends and making informed predictions.
Cumulative frequency is particularly useful in identifying the distribution and tendencies within a dataset.

• It involves summing frequencies from the beginning of the dataset up to a certain point. For example, if you have 26 students who played 0-2 hours, 30 who played 2-4 hours, the cumulative frequency for 4 hours is 26 + 30 = 56.
• In table form, it offers a running total of relative frequency at each interval, providing a clearer picture of how much of the population falls below a certain threshold.

Through cumulative frequency, researchers can glean insights, such as what percentage of students play up to a specific number of hours. This enables visualizing the bulk of activities and facilitates the identification of significant trends and cutoff points within any given dataset.