The masses, mkg, of some parcels are shown below - OCR - GCSE Maths - Question 16 - 2020 - Paper 1

To find the total number of students between 45 and 50 minutes, we calculate the area of the bars in the histogram from 45 to 50 minutes. According to the histogram:

• The frequency density for the interval from 45 to 50 minutes is approximately 1.5.
• The width of this interval is 5 minutes.

Using the formula for area:

$ext{Number of Students} = ext{Frequency Density} imes ext{Width} = 1.5 imes 5 = 7.5$

Similarly, for the interval from 50 to 60 minutes, the frequency density is 5 with the same width:

$ext{Number of Students} = 5 imes 10 = 50$

Adding these gives us:

$7.5 + 50 = 70$

Thus, 70 students took between 45 and 50 minutes to complete the race.

To calculate the mean time, we use the midpoints of each interval in conjunction with the frequency densities:

• For 30 to 40 minutes (Frequency Density = 2), Midpoint = 35: $35 imes 2 imes 10 = 700$
• For 40 to 50 minutes (Frequency Density = 5), Midpoint = 45: $45 imes 5 imes 10 = 2250$
• For 50 to 60 minutes (Frequency Density = 4), Midpoint = 55: $55 imes 4 imes 10 = 2200$
• For 60 to 70 minutes (Frequency Density = 1.5), Midpoint = 65: $65 imes 1.5 imes 10 = 975$

Now, we sum up all the products:

$700 + 2250 + 2200 + 975 = 5815$

Next, we need the total number of students, calculated earlier as 70. Thus, we compute the mean:

ext{Mean} = rac{5815}{70} = 83.0714

Hence, the estimated mean time taken to complete the race is approximately 83.07 minutes.