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9709 P63 - Nov 2010 - Q5
2406
The following histogram illustrates the distribution of times, in minutes, that some students spent taking a shower.
(i) Copy and complete the following frequency table for the data.
Time \( t \) (minutes)
\( 2 < t \le 4 \)
\( 4 < t \le 6 \)
\( 6 < t \le 7 \)
\( 7 < t \le 8 \)
\( 8 < t \le 10 \)
\( 10 < t \le 16 \)
Frequency
(ii) Calculate an estimate of the mean time to take a shower.
Solution
(i) To find the frequencies, use the frequency density from the histogram. The frequency is calculated by multiplying the frequency density by the class width.
For \( 2 < t \le 4 \): Frequency density \(= 10\), Width \(= 2\), Frequency \(= 10 \times 2 = 20\)
For \( 4 < t \le 6 \): Frequency density \(= 22\), Width \(= 2\), Frequency \(= 22 \times 2 = 44\)
For \( 6 < t \le 7 \): Frequency density \(= 34\), Width \(= 1\), Frequency \(= 34 \times 1 = 34\)
For \( 7 < t \le 8 \): Frequency density \(= 30\), Width \(= 1\), Frequency \(= 30 \times 1 = 30\)
For \( 8 < t \le 10 \): Frequency density \(= 15\), Width \(= 2\), Frequency \(= 15 \times 2 = 30\)
For \( 10 < t \le 16 \): Frequency density \(= 6\), Width \(= 6\), Frequency \(= 6 \times 6 = 36\)
(ii) To estimate the mean time, calculate the midpoints of each class interval and use them to find the mean:
