Problem 76 - Precalculus

Question

According to the study of science and engineering indicators by the National Science Foundation (www.nsf.gov), the number of female graduate students in science and cngincering disciplines has increased over the last 30 years. In \(1981,24 \%\) of mathematics graduate students were female and \(23 \%\) of graduate students in computer science were female. In \(1991,32 \%\) of mathematics graduate students and \(21 \%\) of computer science graduate students were female. In \(2001,38 \%\) of mathematics graduate students and \(30 \%\) of computer science graduate students wcre female. Write three \(2 \times 1\) matrices representing the percentage of female graduate students. \(A=\left[\begin{array}{l}\% \text { female }-m a t h-1981 \\ \% \text { female-C.S. }-1981\end{array}\right]\) \(B=\left[\begin{array}{l}\% \text { female-math }-1991 \\ \% \text { female-C.S. }-1991\end{array}\right]\) \(C=\left[\begin{array}{l}\% \text { female-math }-2001 \\ \% \text { female-C.S. }-2001\end{array}\right]\) What does \(C-B\) tell us? What does \(B-A\) tell us? What can you conclude about the number of women pursuing mathematics and computer science graduate degrees? Note: \(\mathbf{C . S .}=\) computer science.

Step-by-Step Solution

Answer

C-B shows a % increase (2001 vs 1991), B-A shows % change (1991 vs 1981). Generally, more women pursued math and CS over 30 years.

Step 1: Create Matrix A for 1981

Matrix A represents the percentage of female graduate students in mathematics and computer science in 1981.\[ A = \begin{bmatrix} 24 \ 23 \end{bmatrix} \]

Step 2: Create Matrix B for 1991

Matrix B represents the percentage of female graduate students in mathematics and computer science in 1991.\[ B = \begin{bmatrix} 32 \ 21 \end{bmatrix} \]

Step 3: Create Matrix C for 2001

Matrix C represents the percentage of female graduate students in mathematics and computer science in 2001.\[ C = \begin{bmatrix} 38 \ 30 \end{bmatrix} \]

Step 4: Calculate C - B

Subtract Matrix B from Matrix C to find the change from 1991 to 2001.\[ C - B = \begin{bmatrix} 38 - 32 \ 30 - 21 \end{bmatrix} = \begin{bmatrix} 6 \ 9 \end{bmatrix} \]This indicates a 6% increase in the proportion of female mathematics graduates and a 9% increase in the proportion of female computer science graduates from 1991 to 2001.

Step 5: Calculate B - A

Subtract Matrix A from Matrix B to find the change from 1981 to 1991.\[ B - A = \begin{bmatrix} 32 - 24 \ 21 - 23 \end{bmatrix} = \begin{bmatrix} 8 \ -2 \end{bmatrix} \]This indicates an 8% increase in the proportion of female mathematics graduates and a 2% decrease in the proportion of female computer science graduates from 1981 to 1991.

Step 6: Conclusion

By comparing these matrices, we can infer the trend over each decade: - From 1981 to 1991, there was an increase in female mathematics graduates but a decrease in female computer science graduates. - From 1991 to 2001, both fields saw an increase, with computer science experiencing a more significant growth in female representation. Overall, there is a trend toward increased female enrollment in graduate programs in these disciplines.

Key Concepts

Matrices in Statistics

Matrices are a fundamental part of statistics and mathematical computations. They provide a structured way to showcase and analyze numerical data. In the context of analyzing statistical data, such as the percentage of female graduate students in STEM fields over several years, matrices allow us to easily perform operations such as addition and subtraction.
For example, consider the given matrices:
- Matrix A for 1981, \[ A = \begin{bmatrix} 24 \ 23 \end{bmatrix} \] represents 24% in mathematics and 23% in computer science for female graduate students.
- Matrix B for 1991, \[ B = \begin{bmatrix} 32 \ 21 \end{bmatrix} \] represents 32% in mathematics and 21% in computer science.
- Finally, Matrix C for 2001, \[ C = \begin{bmatrix} 38 \ 30 \end{bmatrix} \].
By performing operations such as \(C-B\) and \(B-A\), we can determine the changes in percentages over time. This provides valuable insights into trends and shifts within educational demographics.

Female Students in STEM

The presence of female students in STEM (Science, Technology, Engineering, and Mathematics) fields is a critical aspect of ensuring diversity and innovation in these industries. Despite historical underrepresentation, there have been some positive trends over the decades.
From analyzing the matrices:

• From 1981 to 1991, female mathematics graduates increased by 8% while female computer science graduates decreased by 2%.
• From 1991 to 2001, there was growth in both fields—mathematics saw a 6% increase, and computer science saw an impressive 9% increase.

This data suggests that efforts and initiatives during this period may have begun to address gender disparities, especially in attracting more women to study computer science.
Encouraging more female participation in STEM fields is critical as it leads to a more balanced workforce and sparks more diverse creativity and solutions in engineering and technological advancements.

Educational Trends

Examining educational trends, particularly through the lens of gender in STEM fields, can reveal much about societal shifts and priorities over time. The data exemplified by the matrices provides not only a numerical perspective but also reflects the broader efforts made over the years to promote equality and education for all.
Several trends can be highlighted:

• An overall upward trend in female enrollment in graduate programs, particularly noted in mathematics between 1991 and 2001.
• Despite an initial drop in female computer science graduates from 1981-1991, significant gains in subsequent years indicate effective outreach and support programs may have been implemented.

Future educational strategies can build on these trends by employing targeted efforts to further decrease the gender gap in STEM fields, promoting inclusivity, diversity, and parity across all educational and professional areas. This understanding plays a foundational role in shaping educational policies and initiatives today.