Draw an ogive (cumulative frequency curve) on the DIAGRAM SHEET to represent the above data

Here are the solutions for Question 5 and Question 6. Question 5 Data Table Setup: We first set up a table to calculate xf and x^2f. |c|c|c|c|c| Sum (x) & Frequency (f) & xf & x^2 & x^2f \\ 2 & 0 & 0 & 4 & 0 \\ 3 & 3 & 9 & 9 & 27 \\ 4 & 2 & 8 & 16 & 32 \\ 5 & 4 & 20 & 25 & 100 \\ 6 & 4 & 24 & 36 & 144 \\ 7 & 8 & 56 & 49 & 392 \\ 8 & 3 & 24 & 64 & 192 \\ 9 & 2 & 18 & 81 & 162 \\ 10 & 2 & 20 & 100 & 200 \\ 11 & 1 & 11 & 121 & 121 \\ 12 & 1 & 12 & 144 & 144 \\ Total & f = 30 & xf = 222 & & x^2f = 1514 \\ 5.1: Calculate the mean of the data Step 1: Use the formula for the mean of grouped data. x = ( xf)/( f) Step 2: Substitute the values from the table. x = (222)/(30) Step 3: Calculate the mean. x = 7.4 The mean of the data is 7.4. 5.2: Determine the median of the data Step 1: Find the position of the median. The total number of data points is N = 30. The median position is the average of the (N)/(2)-th and ((N)/(2) + 1)-th values. Median position = (30)/(2) = 15-th value and (30)/(2) + 1 = 16-th value So, the median is the average of the 15th and 16th values. Step 2: Create a cumulative frequency column to locate the median values. |c|c|c| Sum (x) & Frequency (f) & Cumulative Frequency (CF) \\ 2 & 0 & 0 \\ 3 & 3 & 3 \\ 4 & 2 & 5 \\ 5 & 4 & 9 \\ 6 & 4 & 13 \\ 7 & 8 & 21 \\ 8 & 3 & 24 \\ 9 & 2 & 26 \\ 10 & 2 & 28 \\ 11 & 1 & 29 \\ 12 & 1 & 30 \\ The 15th value falls in the '7' category (since CF up to 6 is 13, and CF up to 7 is 21). The 16th value also falls in the '7' category. Step 3: Calculate the median. Median = (7 + 7)/(2) = 7 The median of the data is 7. 5.3: Determine the standard deviation of the data Step 1: Use the formula for the population standard deviation for grouped data. = sqrt( (x - x))^2 f f We need to calculate (x - x)^2 f. |c|c|c|c|c| Sum (x) & Frequency (f) & x - x & (x - x)^2 & (x - x)^2 f \\ 2 & 0 & -5.4 & 29.16 & 0 \\ 3 & 3 & -4.4 & 19.36 & 58.08 \\ 4 & 2 & -3.4 & 11.56 & 23.12 \\ 5 & 4 & -2.4 & 5.76 & 23.04 \\ 6 & 4 & -1.4 & 1.96 & 7.84 \\ 7 & 8 & -0.4 & 0.16 & 1.28 \\ 8 & 3 & 0.6 & 0.36 & 1.08 \\ 9 & 2 & 1.6 & 2.56 & 5.12 \\ 10 & 2 & 2.6 & 6.76 & 13.52 \\ 11 & 1 & 3.6 & 12.96 & 12.96 \\ 12 & 1 & 4.6 & 21.16 & 21.16 \\ Total & 30 & & & (x - x)^2 f = 167.2 \\ Step 2: Substitute the sum into the standard deviation formula. = sqrt((167.2)/(30)) Step 3: Calculate the standard deviation. = sqrt(5.57333...) ≈ 2.3608 The standard deviation of the data is approximately 2.36. 5.4: Determine the number of times that the sum of recorded values of the dice is within ONE standard deviation from the mean. Show your calculations. Step 1: Calculate the range within one standard deviation from the mean. Range = (x - , x + ) Range = (7.4 - 2.3608, 7.4 + 2.3608) Range = (5.0392, 9.7608) Step 2: Identify the sum values (x) that fall within this range. The integer sum values within the range (5.0392, 9.7608) are 6, 7, 8, and 9. Step 3: Sum the frequencies for these values. • For x=6, frequency is 4. • For x=7, frequency is 8. • For x=8, frequency is 3. • For x=9, frequency is 2. Total frequency = 4 + 8 + 3 + 2 = 17 The number of times the sum is within one standard deviation from the mean is 17. --- Question 6 Data Table Setup: First, we convert the cumulative frequencies to individual frequencies and find the midpoints for each class. |c|c|c|c|c| Time (hours) & Cumulative Frequency (CF) & Frequency (f) & Midpoint (m) & mf \\ 0 t < 20 & 25 & 25 & 10 & 250 \\ 20 t < 40 & 69 & 69 - 25 = 44 & 30 & 1320 \\ 40 t < 60 & 129 & 129 - 69 = 60 & 50 & 3000 \\ 60 t < 80 & 157 & 157 - 129 = 28 & 70 & 1960 \\ 80 t < 100 & 166 & 166 - 157 = 9 & 90 & 810 \\ 100 t < 120 & 172 & 172 - 166 = 6 & 110 & 660 \\ Total & & f = 172 & & mf = 8000 \\ 6.1: Draw an ogive (cumulative frequency curve) on the DIAGRAM SHEET to represent the above data To draw an ogive: • Plot the upper class boundaries against their corresponding cumulative frequencies. • The points to plot are: (0, 0), (20, 25), (40, 69), (60, 129), (80, 157), (100, 166), (120, 172). • Join these points with a smooth curve. (As an AI, I cannot physically draw the ogive, but these are the steps you would follow on a diagram sheet.) 6.2: Write down the modal class of the data Step 1: Identify the class with the highest frequency from the frequency column calculated above. The frequencies are 25, 44, 60, 28, 9, 6. The highest frequency is 60. Step 2: State the class interval corresponding to the highest frequency. The class interval corresponding to a frequency of 60 is 40 t < 60. The modal class of the data is 40 t < 60. 6.3: Use the ogive (cumulative frequency curve) to estimate the number of learners who watched television more than 80% of the time Step 1: Determine what "80% of the time" refers to. The maximum time in the data is 120 hours. 80\% of 120 hours = 0.80 × 120 = 96 hours We need to estimate the number of learners who watched more than 96 hours. Step 2: Estimate the cumulative frequency at 96 hours using interpolation (as an ogive is not available). The value 96 hours falls in the class 80 t < 100. Cumulative frequency at t=80 is 157. Cumulative frequency at t=100 is 166. The interval width is 100 - 80 = 20. The CF increase is 166 - 157 = 9. CF at 96 = 157 + ((96 - 80)/(100 - 80)) × (166 - 157) CF at 96 = 157 + ((16)/(20)) × 9 CF at 96 = 157 + 0.8 × 9 CF at 96 = 157 + 7.2 = 164.2 So, approximately 164 learners watched 96 hours or less. Step 3: Calculate the number of learners who watched more than 96 hours. Total number of learners is 172. Learners watching > 96 hours = Total learners - CF at 96 hours Learners watching > 96 hours = 172 - 164.2 = 7.8 Since the number of learners must be an integer, we round to the nearest whole number. The estimated number of learners is 8. 6.4: Estimate the mean time (in hours) that learners spent watching television during 3 weeks of the holiday Step 1: Use the formula for the mean of grouped data. x = ( mf)/( f) From the table setup, mf = 8000 and f = 172. Step 2: Substitute the values. x = (8000)/(172) Step 3: Calculate the mean. x ≈ 46.5116 The estimated mean time is approximately 46.51 hours. Send me the next one 📸