Solve according to statistics

Here are the solutions to the problems: Question 1: If the mean of the frequency distribution below is 3, find the value of x. The frequency distribution is: Goals scored (x_i): 1, 2, 3, 4, 5, 6 Number of Matches (f_i): 7, 8, 4, 4, 3, x The formula for the mean of a frequency distribution is: x = ( (x_i · f_i))/( f_i) Step 1: Calculate the sum of (x_i · f_i). (x_i · f_i) = (1 · 7) + (2 · 8) + (3 · 4) + (4 · 4) + (5 · 3) + (6 · x) = 7 + 16 + 12 + 16 + 15 + 6x = 66 + 6x Step 2: Calculate the sum of frequencies ( f_i). f_i = 7 + 8 + 4 + 4 + 3 + x = 26 + x Step 3: Use the given mean (x = 3) to set up the equation and solve for x. 3 = (66 + 6x)/(26 + x) 3(26 + x) = 66 + 6x 78 + 3x = 66 + 6x 78 - 66 = 6x - 3x 12 = 3x x = (12)/(3) x = 4 The value of x is 4. Question 1a: What is the mode? The mode is the value with the highest frequency. With x=4, the frequencies are: Goals scored 1: 7 matches Goals scored 2: 8 matches Goals scored 3: 4 matches Goals scored 4: 4 matches Goals scored 5: 3 matches Goals scored 6: 4 matches (since x=4) The highest frequency is 8, which corresponds to 2 goals scored. The mode is 2. Question 1b: What is the median? First, find the total number of matches (total frequency): N = f_i = 26 + x = 26 + 4 = 30 Since N=30 is an even number, the median is the average of the (N)/(2)-th and ((N)/(2) + 1)-th values. The median will be the average of the 15^th and 16^th values. Let's list the cumulative frequencies: • Goals scored 1: 7 matches (values 1-7 are '1') • Goals scored 2: 7 + 8 = 15 matches (values 8-15 are '2') • Goals scored 3: 15 + 4 = 19 matches (values 16-19 are '3') The 15^th value is 2. The 16^th value is 3. Calculate the median: Median = 15^th value + 16^th value2 = (2 + 3)/(2) = (5)/(2) = 2.5 The median is 2.5. That's 2 down. 3 left today — send the next one.