Question

Three candidates \(X, Y\) and \(Z\) were standing for election to a vacant seat on their college's Student Committee. The members of the electorate (current first-year students, consisting of 150 men and 105 women) were each allowed to cross out the name of the candidate they least wished to be elected, the other two candidates then being credited with one vote each. the following data are known. (a) \(X\) received 100 votes from men, whilst \(Y\) received 65 votes from women. (b) \(Z\) received five more votes from men than \(X\) received from women. (c) The total votes cast for \(X\) and \(Y\) were equal. Analyse this data in such a way that a \(\chi^{2}\) test can be used to determine whether voting was other than random (i) amongst men, and (ii) amongst women.

Short answer

Calculate and compare Chi-squared values separately for men and women to determine voting randomness.

Step-by-step solution

Define Variables and Given Information

Let the votes received by the candidates be represented as follows:\[ X_m = 100 \] (Votes for X from men)\[ Y_w = 65 \] (Votes for Y from women)Let the total number of votes received by X from women be \( X_w \) and by Z from men be \( Z_m \). Given that Z received 5 more votes from men than X received from women:\[ Z_m = X_w + 5 \]Also, let the total number of votes received by X and Y be \( V_X \) and \( V_Y \) where \[ V_X = V_Y \]

Representation of Total Votes

Define the total votes received by the candidates from men and women:\[ 100 + Y_m + Z_m = 2 \times 150 \] (All men voted)\[ 2V_X + Y_m + Z_m = 300 \] (Total votes from men)\[ X_w + 65 + Z_w = 2 \times 105 = 210 \] (All women voted)

Solving System of Equations

From Step 2:\( 100 + Y_m + Z_m = 300 \)Substitute \( Z_m \) with \( X_w + 5 \):\[ 100 + Y_m + X_w + 5 = 300 \]\[ Y_m + X_w = 195 \]For total votes from women:\[ X_w + 65 + Z_w = 210 \]Since total votes are equal:\( V_X = V_Y \)

Calculate Expected and Observed Values

Calculate the observed and expected frequencies separately for men and women:\[ O_m = [100, Y_m, Z_m] \], \[ E_m = [100, \frac {195} {2}, \frac {195} {2}] \]\[ O_w = [X_w, 65, Z_w] \], \[ E_w = [105, 65, 105] \]

Apply Chi-Squared Test

Use Chi-squared formula to compare frequencies:\[ \chi^{2} = \sum \frac { (Observed - Expected)^{2}} {Expected} \] For men and women separately:Calculate \( \chi^{2}_{men} \) and \( \chi^{2}_{women} \)

Conclusion

Compare the results to the critical value to determine if voting was random or not random.

Key concepts

Statistical analysis

Statistical analysis involves collecting and examining data to uncover patterns and trends. In this voting analysis, data from both male and female students were analyzed using statistical techniques to understand their voting behavior.

The basic idea is to make sense of the numbers. We start by defining variables, which help us organize the data in a meaningful way. For example, we count how many people voted for each candidate. This helps break down the bigger problem into smaller, manageable parts.

By understanding how each group voted, we can use statistics to infer if there are any patterns or anomalies in the voting data. This provides a foundation for further statistical tests, like the chi-squared test.

Hypothesis testing

Hypothesis testing is a method used to decide if there is enough evidence to support a certain belief about the data. In this voting analysis, our hypothesis might be that voting is random. If we can prove this wrong, we might find that certain candidates are preferred by specific groups.

Here's how it works: We start with a 'null hypothesis' (usually a statement like 'there is no difference'). We then collect data and use statistical methods to determine if the data supports this hypothesis or if we need to reject it.

In our example, we need to test if votes are distributed randomly among men and women. If the voting was truly random, the distribution of votes should match our expectations based on the numbers we calculated.

Chi-squared test

The chi-squared test is a statistical method used to compare observed data with expected data. It helps us determine if there are significant differences between what we expect and what we actually observe.
This test calculates a 'chi-squared' value using the formula: \[ \chi^{2} = \sum \frac {(Observed - Expected)^{2}}{Expected} \]

In the context of voting patterns, we use this formula to compare the number of votes each candidate received from men and women against the expected values.

For example, if we observe that candidate X received 100 votes from men, and we expect 80 votes based on our calculations, we use the chi-squared formula to see if this difference is significant.

Voting patterns

Voting patterns refer to the way different groups of people vote. By analyzing these patterns, we can understand preferences and behaviors of various groups.

In this exercise, we look at how male and female students vote for different candidates. We compare the actual number of votes each candidate received with the expected number.

For instance, if candidate Y received 65 votes from women, we compare it to what we would expect if the voting were random. This helps us identify any deviations and understand if there's a systematic preference for a particular candidate among male or female voters.

System of equations

A system of equations is a set of mathematical equations that we solve together to find unknown values. In this exercise, we use a system of equations to understand the distribution of votes among candidates.

We use the given data to create equations. For example, if candidate Z received 5 more votes from men than candidate X received from women \[ Z_m = X_w + 5 \], this forms one equation in our system.

By solving these equations, we can find unknown values like the exact number of votes each candidate received from different groups. This forms the basis for our further statistical analysis.