medians and class intervals?

Question

i have a table, it is like this

time, t minutes .......... frequency
0<4 4 8 12 16
a) calculate the class interval in which the median lies

John claims that his results show that the mean time is 10 mins

is he correct? explain briefly you answer

Answer 1

The median is the middle value when they are listed in order.
There are 31 items, so the middle value is the 16th (as there are 15 values on either side of middle value).
Starting with the two values in the class interval 0 to 4, adding the frequencies gives the 16th value lying at the beginning of the class interval 16 to 20.

The mean is the total time divided by 31.
As we don't have the 31 exact times, we have to assume each time is in the middle of its class interval.
We estimate the Total time as 2x2 + 1x6 + 3x10 + 9x14+ 16x18 = 454
So mean = 454/31 = 14.65 which is much bigger than 10.
So it is unlikely that the mean is 10.

BUT we do not know the exact values of the 31 times. The lowest possible mean would be if all these 31 times were at the lowest end of their class intervals.
If this was the case, the Total time would be:
2x0 + 1x4 + 3x8 + 9x12 + 16x16 = 392
So LOWEST possible mean = 392/31 = 12.65
So John cannot be right.

Answer 2

The median is the middle number when they are all written out. So find out how many there are by adding the frequencies
2+1+3+9+16=31 the median is the 16th because then there would be 15 numbers on either side of it. The 16th frequency falls in the class interval 16 The mean erm... erm... erm...
Oh yeah find the midpoint of each group (x)
times it by the frequency (f)
add up all the f times by x (∑fx) (∑ is greek letter sigma meaning add-up)
∑fx divided by ∑f therefore
x f fx
2 2 4
6 1 6
10 3 30
14 9 126
18 16 288
∑fx= 454
∑f= 31
454 divided by 31
equals 14.645161290322580645161290322581
therefore the mean is 14.645161290322580645161290322581
or 14.6

Answer 3

Verify that the bottom fifteen are below 16, and the top 15 are in the range 16
John is wrong, because the mean could range anywhere from 8.3 min to 10.3 min. Why?

No, I am not doing any more than this. Hope I helped!