Calculate the mean, variance, standard deviation, and mode for the given grouped frequency distribution.

Welcome back Ria — missed you this week. To compute the mean, variance, standard deviation, and mode for the given grouped frequency distribution, we first need to determine the class mark (x) for each interval and then calculate the sums required for the formulas. Here is the table with the necessary calculations: | Class Interval | Frequency (f) | Class Mark (x) | fx | x^2 | fx^2 | |:--------------:|:-------------:|:--------------:|:----:|:-----:|:-------:| | 0 - 5 | 9 | 2.5 | 22.5 | 6.25 | 56.25 | | 6 - 11 | 16 | 8.5 | 136.0| 72.25 | 1156.00 | | 12 - 17 | 16 | 14.5 | 232.0| 210.25| 3364.00 | | 18 - 23 | 18 | 20.5 | 369.0| 420.25| 7564.50 | | 24 - 29 | 5 | 26.5 | 132.5| 702.25| 3511.25 | | 30 - 35 | 5 | 32.5 | 162.5| 1056.25| 5281.25 | | 36 - 41 | 1 | 38.5 | 38.5 | 1482.25| 1482.25 | | Total | f = 70 | | fx = 1093.0 | | fx^2 = 22415.5 | a) Mean Step 1: Use the formula for the mean of a grouped frequency distribution. x = ( fx)/( f) Step 2: Substitute the calculated sums. x = (1093)/(70) Step 3: Calculate the mean and round to one decimal place. x = 15.614... ≈ 15.6 The mean is 15.6 minutes. b) Variance Step 1: Use the formula for the variance of a grouped frequency distribution. s^2 = ( fx^2)/( f) - (( fx)/( f))^2 Step 2: Substitute the calculated sums. s^2 = (22415.5)/(70) - ((1093)/(70))^2 Step 3: Calculate the terms. s^2 = 320.2214... - (15.614...)^2 s^2 = 320.2214... - 243.7906... Step 4: Calculate the variance and round to one decimal place. s^2 = 76.4308... ≈ 76.4 The variance is 76.4 minutes^2. c) Standard Deviation Step 1: Use the formula for standard deviation, which is the square root of the variance. s = sqrt(s^2) Step 2: Substitute the calculated variance. s = sqrt(76.4308...) Step 3: Calculate the standard deviation and round to one decimal place. s = 8.742... ≈ 8.7 The standard deviation is 8.7 minutes. d) Mode Step 1: Identify the modal class, which is the class with the highest frequency. The highest frequency is 18, corresponding to the class interval 18 - 23. Step 2: Identify the