The amount of mass loss in kilograms of 200 members was recorded by Amazing Fitness Centre over a one year period. The cumulative frequency curve shows the distribution of the results.

(a)
the median mass loss,
(b)
the interquartile range of the mass loss.
(c)
In order to encourage members to be active in their mass loss, Amazing is waiving a one month membership fee for members who managed to lose at least x kg in a year.

Given that 10% of the members managed to qualify for the waiver, find the value of x.

Answer (c) x = ...
(d)
This box-and-whisker plot represents the distribution of the mass loss of 200 members of another fitness centre, Supreme Fitness Centre.

Make two comments comparing the mass loss of the members in the two fitness centres.

Answer (d) ...
(e)
Amazing Fitness Centre decides to offer "Gold" and "Platinum" membership based on the total mass loss for a year.

Members who lose at least 10 kg but less than 25 kg will be offered "Gold".

Members who lose at least 25 kg will be offered "Platinum".
1. (i)
  A member from Amazing Fitness Centre is chosen at random.
  
  Find the probability that the member selected qualifies for a "Gold" membership.
  
  Answer (e)(i)...
2. (ii)
  Two members from Amazing Fitness Centre are chosen at random.
  
  Andy says that the probability that both members qualify for a "Platinum" membership is $\frac{16}{625}$ .
  
  Explain what he has done wrong and find the correct probability.
  
  Answer (e)(ii)
  
  ...

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Worked solution

6(e)(ii)

1. Identify the number of members who qualify for Platinum from Andy's probability.

Andy’s answer is
$\frac{16}{625}=\left(\frac{4}{25}\right)^2.$
So he used the probability that one member qualifies for Platinum as
$\frac{4}{25}=\frac{32}{200}.$
Thus, there are $32$ Platinum members.

2. Explain the error and calculate the correct probability without replacement.

Andy treated the two selections as independent, but the first selected member is not replaced.

Correct probability:
$P(\text{both Platinum})=\frac{32}{200}\times\frac{31}{199}=\frac{124}{4975}.$

Answer

Andy has assumed the two selections are independent, as if the first member is replaced. Since two members are chosen from the 200 members without replacement, the correct probability is
$\frac{32}{200}\times\frac{31}{199}=\frac{124}{4975}.$

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The amount of mass loss in kilograms of 200 members was recorded by Amazing Fitness Centre over a one year period. The cumulative frequency curve shows the distribution of the results.

Worked solution

6(e)(ii)

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