Question

a) what is the interquartile range of the lengths of bananas?
b) a shop only buys bananas with lengths between 13 cm and 20 cm. estimate the fraction of the 80 bananas that the shop buys.
cumulative frequency graph titled lengths of 80 bananas with x - axis length (cm) from 0 to 30 and y - axis cumulative frequency from 0 to 80.

Explanation:

Part (a)

Step1: Find \( Q_1 \) (Lower Quartile)

The total number of bananas \( n = 80 \). For the lower quartile, we find the value at \( \frac{n}{4}=\frac{80}{4} = 20 \)th cumulative frequency. From the cumulative frequency graph, when cumulative frequency is 20, the length (x - value) is approximately 12 cm (by reading the graph: find y = 20, move horizontally to the curve, then down to x - axis).

Step2: Find \( Q_3 \) (Upper Quartile)

For the upper quartile, we find the value at \( \frac{3n}{4}=\frac{3\times80}{4}=60 \)th cumulative frequency. From the cumulative frequency graph, when cumulative frequency is 60, the length (x - value) is approximately 20 cm (find y = 60, move horizontally to the curve, then down to x - axis).

Step3: Calculate Interquartile Range (IQR)

The interquartile range is \( IQR=Q_3 - Q_1 \). Substituting the values of \( Q_3 = 20 \) cm and \( Q_1 = 12 \) cm, we get \( IQR=20 - 12=8 \) cm.

Step1: Find cumulative frequency at 13 cm

First, we find the cumulative frequency when the length is 13 cm. From the cumulative frequency graph, we estimate the cumulative frequency (y - value) corresponding to \( x = 13 \) cm. By looking at the graph, when \( x = 13 \), the cumulative frequency \( CF_{13}\approx10 \) (approximate reading: move up from x = 13 to the curve, then left to y - axis).

Step2: Find cumulative frequency at 20 cm

Next, we find the cumulative frequency when the length is 20 cm. From the cumulative frequency graph, when \( x = 20 \) cm, the cumulative frequency \( CF_{20}\approx60 \) (move up from x = 20 to the curve, then left to y - axis).

Step3: Calculate the number of bananas in the range

The number of bananas with length between 13 cm and 20 cm is \( CF_{20}-CF_{13}=60 - 10 = 50 \).

Step4: Calculate the fraction

The total number of bananas is 80. So the fraction is \( \frac{50}{80}=\frac{5}{8} \).

Answer:

The interquartile range is \( \boldsymbol{8} \) cm.

Part (b)