Probability of Preference for Mascot A

Oct 13, 2025 by miodeya.com 31 views

Introduction

In a recent survey conducted by student representatives, classmates were asked to express their preference for a school mascot for a new school. The results of the survey are presented in the table below. In this analysis, we will examine the data to determine which pair of samples seems most representative of student preferences.

Survey Results

Sample	Sample Size	Mascot A	Mascot B	Mascot C	Mascot D
A	100	60	20	10	10
B	120	40	30	20	30
C	150	30	40	20	10
D	180	20	40	30	10

Mathematical Analysis

To determine which pair of samples seems most representative of student preferences, we need to analyze the data using mathematical concepts. We will use the concept of probability and statistical analysis to compare the samples.

Probability of Preference

Let's assume that the probability of a student preferring Mascot A is P(A), the probability of a student preferring Mascot B is P(B), and so on. We can calculate the probability of a student preferring each mascot using the sample sizes and the number of students who prefer each mascot.

Probability of Preference for Mascot A

P(A) = (Number of students who prefer Mascot A) / (Total sample size) = (60 + 40 + 30 + 20) / (100 + 120 + 150 + 180) = 150 / 550 = 0.273

Probability of Preference for Mascot B

P(B) = (Number of students who prefer Mascot B) / (Total sample size) = (20 + 30 + 40 + 40) / (100 + 120 + 150 + 180) = 130 / 550 = 0.236

Probability of Preference for Mascot C

P(C) = (Number of students who prefer Mascot C) / (Total sample size) = (10 + 20 + 20 + 30) / (100 + 120 + 150 + 180) = 80 / 550 = 0.145

Probability of Preference for Mascot D

P(D) = (Number of students who prefer Mascot D) / (Total sample size) = (10 + 30 + 10 + 10) / (100 + 120 + 150 + 180) = 60 / 550 = 0.109

Statistical Analysis

To compare the samples, we can use statistical tests such as the chi-squared test or the t-test. However, since we are dealing with categorical data, we can use the chi-squared test.

Chi-Squared Test

The chi-squared test is used to determine whether there is a significant difference between the observed frequencies and the expected frequencies. The test statistic is calculated as follows:

χ² = Σ [(observed frequency - expected frequency)² / expected frequency]

We can calculate the expected frequencies using the probability of preference for each mascot.

Expected Frequencies

Expected frequency for Mascot A = P(A) × Total sample size = 0.273 × 550 = 150

Expected frequency for Mascot B = P(B) × Total sample size = 0.236 × 550 = 130

Expected frequency for Mascot C = P(C) × Total sample size = 0.145 × 550 = 80

Expected frequency for Mascot D = P(D) × Total sample size = 0.109 × 550 = 60

Chi-Squared Test Statistic

χ² = [(60 - 150)² / 150] + [(20 - 130)² / 130] + [(10 - 80)² / 80] + [(10 - 60)² / 60] = 0.36 + 0.46 + 0.56 + 0.36 = 1.94

Degrees of Freedom

The degrees of freedom for the chi-squared test is calculated as follows:

df = (number of categories - 1) = (4 - 1) = 3

P-Value

The p-value is calculated as follows:

p-value = P(χ² > χ² observed) = P(χ² > 1.94) = 0.60

Conclusion

Based on the chi-squared test, we can conclude that there is no significant difference between the observed frequencies and the expected frequencies. This suggests that the samples are representative of student preferences.

Recommendation

Based on the analysis, we recommend that the school mascot be Mascot A, as it has the highest probability of preference (0.273). However, it is essential to note that the results are based on a small sample size and may not be representative of the entire student population.

Limitations

The analysis has several limitations. Firstly, the sample size is small, which may not be representative of the entire student population. Secondly, the survey was conducted by student representatives, which may have introduced bias. Finally, the analysis assumes that the probability of preference for each mascot is independent, which may not be the case.

Future Research

Q: What is the purpose of the survey?

A: The purpose of the survey is to determine which school mascot is most preferred by students.

Q: How was the survey conducted?

A: The survey was conducted by student representatives who asked their classmates to express their preference for a school mascot.

Q: What are the possible school mascots?

A: The possible school mascots are Mascot A, Mascot B, Mascot C, and Mascot D.

Q: What are the results of the survey?

A: The results of the survey are presented in the table below:

Sample	Sample Size	Mascot A	Mascot B	Mascot C	Mascot D
A	100	60	20	10	10
B	120	40	30	20	30
C	150	30	40	20	10
D	180	20	40	30	10

Q: How was the probability of preference calculated?

A: The probability of preference was calculated by dividing the number of students who prefer each mascot by the total sample size.

Q: What is the chi-squared test?

A: The chi-squared test is a statistical test used to determine whether there is a significant difference between the observed frequencies and the expected frequencies.

Q: What is the p-value?

A: The p-value is the probability of observing a test statistic at least as extreme as the one observed, assuming that the null hypothesis is true.

Q: What is the conclusion of the analysis?

A: The conclusion of the analysis is that there is no significant difference between the observed frequencies and the expected frequencies, suggesting that the samples are representative of student preferences.

Q: What is the recommendation of the analysis?

A: The recommendation of the analysis is that the school mascot be Mascot A, as it has the highest probability of preference (0.273).

Q: What are the limitations of the analysis?

A: The limitations of the analysis are that the sample size is small, which may not be representative of the entire student population, and that the survey was conducted by student representatives, which may have introduced bias.

Q: What are the future research directions?

A: The future research directions are to increase the sample size, reduce bias, and consider other factors that may influence student preferences, such as age, gender, and academic performance.

Q: How can the results of the survey be used?

A: The results of the survey can be used to inform the decision-making process for selecting a school mascot. The results can also be used to identify areas for improvement in the survey process and to develop more effective strategies for engaging students in the decision-making process.

Q: What are the implications of the analysis for the school community?

A: The implications of the analysis for the school community are that the school mascot should be selected based on the preferences of the students, and that the school should strive to create a sense of community and inclusivity among students, faculty, and staff.