Question

The administration is considering a total ban on student automobiles. You have conducted a poll on this issue of 20 fellow students and 20 of the neighbors who live around the campus and have calculated scores for your respondents. On the scale you used, a high score indicates strong opposition to the proposed ban. The scores are presented here for both groups. Calculate an appropriate measure of central tendency and compare the two groups in a sentence or two. $$\begin{array}{crc}\text{ Students }& & \text{ Neighbors }\\ 10& 11& 0& 7\\ 10& 9& 1& 6\\ 10& 8& 0& 0\\ 10& 11& 1& 2\\ 9& 8& 7& 4\\ 10& 11& 11& 0\\ 9& 7& 0& 0\\ 5& 1& 1& 10\\ 5& 2& 10& 9\\ 0& 10& 10& 0\end{array}$$

Short answer

Students have a higher mean score (7.8) compared to neighbors (5.1). The median score for students is also higher (9) compared to neighbors (4.5).

Step-by-step solution

Organize the Data

Separate the scores into two lists: one for students and one for neighbors.Students: [10, 10, 10, 10, 9, 10, 9, 5, 5, 0]Neighbors: [11, 9, 8, 11, 8, 11, 7, 1, 2, 10]

Calculate the Mean Score

Calculate the average score for each group.Students: $\text{Mean}=\frac{10+10+10+10+9+10+9+5+5+0}{10}=\frac{78}{10}=7.8$Neighbors: $\text{Mean}=\frac{11+9+0+1+7+11+0+2+0+10}{10}=\frac{51}{10}=5.1$

Calculate the Median Score

Find the median score for each group by ordering the numbers and picking the middle one.Students:- Sorted scores: [0, 5, 5, 9, 9, 10, 10, 10, 10, 10]- Median: (9 + 9) / 2 = 9Neighbors:- Sorted scores: [0, 0, 0, 1, 2, 7, 9, 10, 11, 11]- Median: (2 + 7) / 2 = 4.5

Compare the Measures of Central Tendency

Compare the means and medians of the two groups in one or two sentences.The average score for students is higher than for neighbors (7.8 vs. 5.1). The medians show a similar trend with students having a median score of 9 compared to 4.5 for neighbors.

Key concepts

Mean Calculation

A central tendency measure helps summarize a set of data with a single value. The mean or average is a crucial example of such measures. Here’s how you can calculate it:

1. **Add all the numbers in your data set**. For example, the student survey scores are: [10, 10, 10, 10, 9, 10, 9, 5, 5, 0].
2. **Divide the total by the number of values**. For students, the total score is 78, and there are 10 students. So, the mean score would be: $$\text{Mean}=\frac{78}{10}=7.8$$

Similarly, for neighbors, their scores are: [11, 9, 0, 1, 7, 11, 0, 2, 0, 10]. The total score is 51, and there are 10 neighbors:
$$\text{Mean}=\frac{51}{10}=5.1$$

The mean score shows the average attitude towards the student automobile ban. Students have a higher average score, suggesting greater opposition compared to neighbors.

Median Calculation

The median is another measure of central tendency which shows the middle value in an ordered data set.

To find the median:
1. **Arrange the numbers in order**. For students, this becomes: [0, 5, 5, 9, 9, 10, 10, 10, 10, 10].
2. **Find the middle number**. If there’s an even number of values, calculate the average of the two middle numbers. For students, the median is: $$\text{Median}=\frac{9+9}{2}=9$$

For neighbors, the sorted set is: [0, 0, 0, 1, 2, 7, 9, 10, 11, 11]. Here the median is:
$$\text{Median}=\frac{2+7}{2}=4.5$$

The median gives a better sense of the typical value especially when a data set contains outliers. For this survey, the median shows that students generally have stronger opposition to the ban compared to neighbors.

Data Analysis

Data analysis involves systematically applying statistical methods to describe and interpret data.

1. **Organize your data**: Separate the scores into groups like 'students' and 'neighbors'.
2. **Calculate necessary measures**: Here it means finding the mean and median for each group.
3. **Interpret the results**: The average (mean) and middle value (median) for students are both higher than for neighbors, indicating stronger opposition to the ban among students.

Through such analysis, you can better understand the underlying trends and sentiments represented by raw data. The method revealed that students are more opposed to the proposed ban than the neighboring community.

Student Survey

Conducting a student survey helps gather opinions on important issues within a school community. Steps include:

1. **Create a questionnaire**: Formulate questions that measure attitudes or opinions. In this case, a question about the students' stance on the automobile ban was used.
2. **Distribute the survey**: Get responses from a representative group. Twenty students took part in this exercise.
3. **Analyze the results**: Calculate measures like the mean and median scores. For students, the mean score was 7.8, and the median was 9, showing significant opposition to the ban.

Using surveys and these calculations allow schools to make informed decisions based on the views of their student bodies.

Neighbor Survey

Surveys aren't just for students; involving the neighboring community provides a broader perspective. Consider the following steps:

1. **Design the survey**: Like for students, create targeted questions to understand neighbors' views.
2. **Collect responses**: In our example, 20 neighbors participated. Their scores were noted and helped gauge opposition or support.
3. **Analyze and compare**: Neighbors had a mean score of 5.1 and a median of 4.5, indicating less opposition compared to students.

This comparison between surveys from different groups can highlight differing community perspectives. In this case, it helped show that neighbors are less opposed to the proposed automobile ban than students.