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.
ScanSolve AI Answer
Here's how to solve the problem: a) Find the value of a. Step 1: Write the formula for the mean of grouped data and calculate the sum of (frequency × score) and the sum of frequencies. The mean is given by the formula: Mean = ( (f × x))/( f) From the table: Scores (x): 5, 6, 7, 8, 9, 10 Frequencies (f): 2, a, 4, 1, 2, 1 Calculate (f × x): (5 × 2) + (6 × a) + (7 × 4) + (8 × 1) + (9 × 2) + (10 × 1) = 10 + 6a + 28 + 8 + 18 + 10 = 74 + 6a Calculate f: 2 + a + 4 + 1 + 2 + 1 = 10 + a Step 2: Substitute the given mean and the sums into the formula and solve for a. Given that the mean is 7: 7 = (74 + 6a)/(10 + a) Multiply both sides by (10 + a): 7(10 + a) = 74 + 6a 70 + 7a = 74 + 6a Subtract 6a from both sides: 70 + a = 74 Subtract 70 from both sides: a = 74 - 70 a = 4 The value of a is 4. b) Find the mode and median. Step 1: Update the frequencies with the value of a=4. Scores (x): 5, 6, 7, 8, 9, 10 Frequencies (f): 2, 4, 4, 1, 2, 1 Total number of learners ( f) = 2 + 4 + 4 + 1 + 2 + 1 = 14. Step 2: Determine the mode. The mode is the score with the highest frequency. The highest frequency in the updated data is 4, which corresponds to scores 6 and 7. The mode is 6 and 7. Step 3: Determine the median. The median is the middle value when the data is ordered. Since there are 14 learners (an even number), the median is the average of the ((14)/(2))^th and ((14)/(2) + 1)^th values, which are the 7^th and 8^th values. Let's find the cumulative frequencies: • Scores of 5: 2 learners (1st to 2nd position) • Scores of 6: 4 learners (3rd to 6th position) • Scores of 7: 4 learners (7th to 10th position) The 7^th value is 7, and the 8^th value is 7. Median = (7 + 7)/(2) = (14)/(2) = 7 The median is 7. c) Find the probability of selecting a learner who scored the mode marks. Step 1: Identify the mode marks and the number of learners who scored them. From part b), the mode marks are 6 and 7. Number of learners who scored 6 marks = 4. Number of learners who scored 7 marks = 4. Total number of learners who scored mode marks = 4 + 4 = 8. Step 2: Calculate the probability. The total number of learners is 14. The probability of selecting a learner who scored the mode marks is: P(mode marks) = Number of learners with mode marksTotal number of learners P(mode marks) = (8)/(14) Simplify the fraction: P(mode marks) = (4)/(7) The probability is (4)/(7). Send me the next one 📸