1. What is the purpose of the Gauss-Jordon elimination method?
a. Solving systems of linear equations
b. Finding determinants of matrices
c. Calculating eigenvalues
d. Simplifying matrices
2. What is Cramer's Rule used for?
a. Solving systems of linear equations
b. Finding determinants of matrices
c. Calculating eigenvalues
d. Simplifying matrices
3. Which of the following is an example of descriptive statistics?
a. Calculating a z-score
b. Finding the correlation coefficient
c. Calculating the mean and standard deviation
d. Conducting a hypothesis test
4. What is inferential statistics used for?
a. Summarizing data
b. Making predictions or drawing conclusions about a population based on sample data
c. Describing the relationship between two variables
d. Testing hypotheses
5. What does the regression line represent in least squares regression?
a. The line that passes through the mean of the data points
b. The line that minimizes the sum of the squared differences between observed and predicted values
c. The line with the highest correlation coefficient
d. The line that best fits the data points
Application (40 marks)
1. Solve the following system of linear equations using the Gauss-Jordon elimination method:
2x + 3y = 7
4x - y = 2
2. Use Cramer's Rule to solve the following system of equations:
3x + 2y = 6
x - 4y = -2
3. Calculate the simple interest on a loan of $500 at a rate of 3% for 2 years.
4. Perform a least squares regression analysis on the following dataset and calculate the equation of the regression line:
x: 1, 2, 3, 4, 5
y: 3, 5, 7, 9, 11
5. Calculate the mean and standard deviation of the following dataset:
10, 12, 15, 18, 22
(Note: Make sure to show all your calculations and provide clear explanations for each question.)
1. The purpose of the Gauss-Jordon elimination method is to solve systems of linear equations.
2. Cramer's Rule is used to solve systems of linear equations.
3. For a loan of $500 at a rate of 3% for 2 years, the simple interest can be calculated as follows:
Simple Interest = Principal x Rate x Time
Simple Interest = $500 x 0.03 x 2 = $30
4. To perform a least squares regression analysis on the dataset provided:
First, calculate the mean of both x and y:
Mean of x = (1+2+3+4+5)/5 = 3
Mean of y = (3+5+7+9+11)/5 = 7
Then, calculate the slope of the regression line:
Slope (b) = Σ[(x_i - x_mean)(y_i - y_mean)] / Σ[(x_i - x_mean)^2]
Slope (b) = (1-3)(3-7) + (2-3)(5-7) + (3-3)(7-7) + (4-3)(9-7) + (5-3)(11-7) / (1-3)^2 + (2-3)^2 + (3-3)^2 + (4-3)^2 + (5-3)^2
Slope (b) = -8 / 10 = -0.8
Finally, calculate the intercept of the regression line:
Intercept (a) = y_mean - b * x_mean
Intercept (a) = 7 - (-0.8 * 3) = 9.4
Therefore, the equation of the regression line is y = -0.8x + 9.4
5. To calculate the mean and standard deviation of the dataset {10, 12, 15, 18, 22}:
Mean = (10 + 12 + 15 + 18 + 22) / 5 = 15.4
Standard Deviation = √[Σ(x_i - mean)^2 / (n-1)]
Standard Deviation = √[(10-15.4)^2 + (12-15.4)^2 + (15-15.4)^2 + (18-15.4)^2 + (22-15.4)^2 / (5-1)]
Standard Deviation = √[27.04 + 11.56 + 0.16 + 6.76 + 42.24 / 4]
Standard Deviation = √87.76 / 4 = √21.94 = 4.68
Therefore, the mean of the dataset is 15.4 and the standard deviation is 4.68.