This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here's how to solve the problem: Part a) Find the value of x and y. Step 1: Set up equations based on the given information. The table shows the number of letters (x) and the number of words (f). The total number of words is 50, so f = 50. The average number of letters per word (mean) is 5, so x = 5. From the sum of frequencies: 1 + 3 + x + 10 + 12 + y + 6 + 4 + 1 = 50 37 + x + y = 50 x + y = 13 (Equation 1) From the mean formula, x = ( fx)/( f): fx = (1 × 1) + (2 × 3) + (3 × x) + (4 × 10) + (5 × 12) + (6 × y) + (7 × 6) + (8 × 4) + (9 × 1) fx = 1 + 6 + 3x + 40 + 60 + 6y + 42 + 32 + 9 fx = 190 + 3x + 6y Since x = 5 and f = 50: 5 = (190 + 3x + 6y)/(50) 250 = 190 + 3x + 6y 60 = 3x + 6y Divide by 3: 20 = x + 2y (Equation 2) Step 2: Solve the system of equations. We have: 1) x + y = 13 2) x + 2y = 20 Subtract Equation 1 from Equation 2: (x + 2y) - (x + y) = 20 - 13 y = 7 Substitute y=7 into Equation 1: x + 7 = 13 x = 13 - 7 x = 6 The values are x=6, y=7. Part b) Calculate the standard deviation of the distribution. Step 3: Complete the frequency table and calculate fx^2. Now that we know x=6 and y=7, the distribution is: | Number of letters (x) | Number of words (f) | fx | x^2 | fx^2 | | :-------------------- | :------------------ | :-- | :---- | :----- | | 1 | 1 | 1 | 1 | 1 | | 2 | 3 | 6 | 4 | 12 | | 3 | 6 | 18 | 9 | 54 | | 4 | 10 | 40 | 16 | 160 | | 5 | 12 | 60 | 25 | 300 | | 6 | 7 | 42 | 36 | 252 | | 7 | 6 | 42 | 49 | 294 | | 8 | 4 | 32 | 64 | 256 | | 9 | 1 | 9 | 81 | 81 | | Total | 50 | 250 | | 1410 | From the table, f = 50, fx = 250, and fx^2 = 1410. Step 4: Calculate the variance (^2). The formula for variance is: ^2 = ( fx^2)/( f) - x^2 We know x = 5. ^2 = (1410)/(50) - (5)^2 ^2 = 28.2 - 25 ^2 = 3.2 Step 5: Calculate the standard deviation (). The standard deviation is the square root of the variance: = sqrt(3.2) ≈ 1.78885... Rounding to two decimal places: ≈ 1.79 The standard deviation is 1.79. 3 done, 2 left today. You're making progress.