The masses of 40 oranges were measured. The cumulative frequency curve below shows the distribution of the masses.

((a))
Use the two diagrams to complete this table for the two types of fruits.
1. Type Lower quartile Median Upper quartile Inter-quartile range
  Orange g g g 11 g
  Apple g 126 g g 11 g
((b))
Describe how the cumulative frequency curve for the apples may differ from the curve for the oranges.
((c))
Below are two statements comparing the distributions of the masses of oranges and apples.

For each statement, write True or False. Give a reason for each answer, stating clearly which statistics you use to make your decision.
1. ((i))
  The apples are heavier than the oranges.
  
  ... because ...
  
  ... because ...
2. ((ii))
  A greater percentage of oranges weigh more than 131g than apples.
  
  ... because ...
  
  ... because ...
((d))
The grouped frequency table for the masses of the oranges is given below.
1. ((i))
  Calculate an estimate of the mean mass.
  
  Answer ... g
2. ((ii))
  Calculate an estimate of the standard deviation.
  
  Answer ... g
3. ((iii))
  2 oranges are chosen at random without replacement.
  Calculate the probability that at least one of the oranges weigh at least 125 g.
  
  Answer ...

Type	Lower quartile	Median	Upper quartile	Inter-quartile range
Orange	g	g	g	11 g
Apple	g	126 g	g	11 g

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Worked solution

1. Find the class midpoints and compute the estimated total mass.

Class midpoints: $110,120,130,140$

$\sum fx=5(110)+10(120)+20(130)+5(140)=5050$

2. Divide by the total frequency to estimate the mean.

Total frequency $=40$

$\text{Estimated mean}=\frac{5050}{40}=126.25$

Answer

$125$ g

1. Use class midpoints and the estimated mean to calculate the estimated standard deviation.

Class midpoints: $110,120,130,140$ .

Using the estimated mean from part (d)(i), $\bar{x}=125$ ,

\begin{aligned} \text{standard deviation} &\approx \sqrt{\frac{5(110-125)^2+10(120-125)^2+20(130-125)^2+5(140-125)^2}{40}}\\ &=\sqrt{\frac{5(225)+10(25)+20(25)+5(225)}{40}}\\ &=\sqrt{\frac{3000}{40}}\\ &=\sqrt{75}\\ &=8.66\text{ g} \end{aligned}

Answer

$8.66$ g

1. Find the number of oranges weighing at least 125 g and less than 125 g.

From the table,

Number weighing at least $125$ g $=20+5=25$ .

Number weighing less than $125$ g $=5+10=15$ .

2. Use the complement of 'at least one' and calculate the probability without replacement.

$P(\text{at least one weighs at least }125\text{ g})$

$=1-P(\text{both weigh less than }125\text{ g})$

$=1-\frac{15}{40}\times\frac{14}{39}$

$=1-\frac{7}{52}=\frac{45}{52}$

Answer

$\dfrac{11}{16}$

AI analysis — may contain mistakes.

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