Question

Your teacher lets the students decide whether to have their test on Friday or Monday. The table shows the results from four surveys of randomly selected students in your grade who are taking the same class. The students are asked whether they want to have the test on Friday. $$ \begin{array}{|c|c|c|} \hline \begin{array}{c} \text { Sample } \\ \text { Size } \end{array} & \begin{array}{c} \text { Number of } \\ \text { "Yes" Responses } \end{array} & \begin{array}{c} \text { Percent of } \\ \text { Votes } \end{array} \\ \hline 10 & 8 & 80 \% \\ 20 & 12 & 60 \% \\ 30 & 16 & 53.3 \% \\ 40 & 18 & 45 \% \\ \hline \end{array} $$ a. Based on the results of the first two surveys, do you think the test will be on Friday? Explain. b. Based on the results in the table, do you think the test will be on Friday? Explain.

Short answer

a. Based on the results of the first two surveys, it initially seems like the test will be held on Friday. However, with the decreasing trend noticeable as the sample size increases, it is difficult to definitively say that the test will indeed be on Friday. | b. When all the survey results are considered, the percentage of 'Yes' responses decreases with larger sample size. Therefore, while the initial surveys would suggest the test would be on Friday, examining all data suggests otherwise. The test might not be on Friday if the same declining trend continues with a larger group of students.

Step-by-step solution

Analyze the first two surveys

First, look at the results from the first two surveys. In the first survey with 10 students, 80% or 8 students said 'Yes'. In the second survey, with 20 students, 60% or 12 students said 'Yes'. The results from the first two surveys shows a majority in favor of having the test on Friday.

Identify the trend

Identify the pattern from the results of the first two surveys: as the sample size increases, the percentage of 'Yes' responses decreases. If the trend continues, it indicates that as more students are surveyed, the majority may shift away from 'Yes'

Analyze the remaining surveys

Now analyze the results of the remaining surveys. For the survey with 30 students, 53.3% or roughly 16 students said 'Yes'. For the last survey with 40 students, 45% or 18 students said 'Yes'. The decreasing trend in the percentage of 'Yes' responses continues with the larger sample sizes.

Draw a conclusion

Taking all the survey results into account, it can be observed that as the sample size increases, the percentage of students who want the test on Friday decreases. This suggests that if more students were surveyed, the trend might continue, resulting in less than 50% of the students wanting the test on Friday.

Key concepts

Survey Results Interpretation

Understanding survey results is crucial for making informed decisions. In the scenario of deciding the test day, the survey results can reveal students' preferences. Here, we have surveys with different sample sizes. Each sample provides a snapshot of student opinion at that moment.

When interpreting these results, note the percentage of 'Yes' responses. It tells us how many students prefer the test on Friday. Initially, smaller groups (Sample 10 and 20) indicated a majority in favor of Friday, with 80% and 60% voting 'Yes' respectively.

However, interpreting survey results should consider not only this majority but also how results might change with larger groups. This leads us to analyze possible trends and patterns that can emerge.

Trend Identification

Identifying trends helps make predictions based on data patterns. With the survey data at hand, we notice a clear pattern: as the sample size increases, the percentage of students voting 'Yes' decreases.

Initially, 80% of a small sample supported having the test on Friday, but this percentage drops significantly as the sample size grows to 40 students. By analyzing these trends, we can hypothesize that with even more data collection, the preference for a Friday test might decline further.

Spotting such trends is vital because it helps us understand how opinions might shift as more students voice their preferences, leading to more balanced decision-making.

Sample Size Effect

Sample size plays a key role in the accuracy of survey results. A small sample may not reliably represent the entire population's views.

In our scenario, the survey with just 10 students showed 80% of them wanting a Friday test. But as the sample sizes increased to 20, 30, and finally 40 students, the percentage of 'Yes' votes dropped to 45%.

This shows a crucial point: larger samples can provide more reliable data that may differ significantly from what smaller samples suggest. Understanding the effect of sample size ensures we don’t make misleading conclusions based on insufficient data.

Percent Calculation

Calculating percentages is an essential skill in analyzing survey results. It helps quantify responses succinctly.

Let's break down how this is applied in the surveys. For each sample size, the number of 'Yes' responses is divided by the total sample size and then multiplied by 100. For example, in the survey of 10 students, 8 want the test on Friday. So, the percentage is calculated as \[ \frac{8}{10} \times 100 = 80\% \].

Similarly, calculations for other surveys reveal a decreasing percentage of 'Yes' votes. By understanding percent calculation, you can easily summarize data for quick insights, facilitating better decision-making processes.