Math

How do you calculate the interquartile range (IQR) of a data set and why is it important in determining data variability?To calculate the interquartile range (IQR) of a data set, first arrange the data in ascending order. Divide the data into four equal parts to find the first quartile (Q1) and the third quartile (Q3). The IQR is the difference between Q3 and Q1 (IQR = Q3 – Q1). The IQR is important because it measures the spread of the middle 50% of the data, providing a robust measure of variability that is less affected by outliers compared to the range.

What is the formula for calculating the area of a triangle?The area of a triangle can be calculated using the formula: Area = 0.5 * base * height. Alternatively, for a triangle with sides of lengths a, b, and c, and semi-perimeter s = (a + b + c) / 2, Heron’s formula can be used: Area = sqrt(s * (s – a) * (s – b) * (s – c)).

How do you solve the equation 2x^2 + 3x – 5 = 0 using the quadratic formula?To solve the quadratic equation 2x^2 + 3x – 5 = 0 using the quadratic formula, we use the formula x = [-b ± sqrt(b^2 – 4ac)] / 2a. Here, a = 2, b = 3, and c = -5. Plugging in these values, we get x = [-3 ± sqrt(3^2 – 4*2*(-5))] / (2*2). Simplifying, we find x = [-3 ± sqrt(9 + 40)] / 4, or x = [-3 ± sqrt(49)] / 4. Therefore, x = [-3 ± 7] / 4, giving us two solutions: x = 1 and x = -5/2.

What are the techniques for evaluating definite integrals using the Fundamental Theorem of Calculus?The Fundamental Theorem of Calculus connects differentiation and integration, providing a method to evaluate definite integrals. To use it, find the antiderivative F(x) of the integrand f(x). Then, evaluate F(b) – F(a), where [a, b] is the interval of integration. This result gives the net area under the curve f(x) from a to b.

What is the solution to the equation 3x + 5 = 20?To solve the equation 3x + 5 = 20, first subtract 5 from both sides to get 3x = 15. Then, divide both sides by 3 to find x = 5. Thus, the solution is x = 5.

What are the basic steps for adding and subtracting fractions with different denominators?To add or subtract fractions with different denominators, follow these steps: 1) Find the least common denominator (LCD) of the fractions. 2) Convert each fraction to an equivalent fraction with the LCD. 3) Add or subtract the numerators, keeping the denominator the same. 4) Simplify the resulting fraction if possible.

How do you solve a linear equation in one variable?To solve a linear equation in one variable, isolate the variable on one side of the equation using inverse operations. Simplify both sides of the equation as needed by combining like terms and performing arithmetic operations. The solution is the value of the variable that makes the equation true.

What are the basic steps to solve a linear equation?To solve a linear equation, follow these steps: 1) Simplify both sides of the equation by combining like terms. 2) Move variable terms to one side and constant terms to the other using addition or subtraction. 3) Isolate the variable by dividing or multiplying both sides by the coefficient of the variable.

How do you solve for x in the equation 5x + 12 = 37?To solve for x in the equation 5x + 12 = 37, first subtract 12 from both sides to get 5x = 25. Then, divide both sides by 5 to find x = 5.

How can you use a chi-square test to determine whether there is a significant association between categorical variables in a contingency table?To use a chi-square test to determine if there is a significant association between categorical variables in a contingency table, follow these steps: (1) Formulate the null and alternative hypotheses. (2) Calculate the expected frequencies for each cell. (3) Compute the chi-square statistic using the formula χ² = Σ[(O-E)²/E], where O is the observed frequency and E is the expected frequency. (4) Determine the degrees of freedom (df) as (rows-1)*(columns-1). (5) Compare the chi-square statistic to the critical value from the chi-square distribution table at the desired significance level. If the calculated χ² exceeds the critical value, reject the null hypothesis, indicating a significant association.