A tyre manufacturing company researched into the... | WAEC SSCE - Further Mathematics - 2011 (Essay)

Question 1 Report

A tyre manufacturing company researched into the life span of one type of their motorcycle tyres. The results were as follows :

Distance (100km)	10-19	20-29	30-39	40-49	50-59	60-69
Number of tyres	30	69	93	57	36	15

(a) Draw a histogram for the distribution.

(b) Use the histogram to estimate the mode.

Answer

(a) Histogram

Since all the class intervals have the same width, namely 10 (in units of 100 km), the bar heights are proportional to the frequencies. Use continuous class boundaries as follows:

Distance (100 km)	Class boundaries	Frequency
10–19	9.5–19.5	30
20–29	19.5–29.5	69
30–39	29.5–39.5	93
40–49	39.5–49.5	57
50–59	49.5–59.5	36
60–69	59.5–69.5	15

graph — Histogram with contiguous equal-width bars: 9.5–19.5, 19.5–29.5, 29.5–39.5, 39.5–49.5, 49.5–59.5 and 59.5–69.5; corresponding heights are 30, 69, 93, 57, 36 and 15.

(b) Estimate of the mode

The modal class is \(30\text{–}39\), with boundaries \(29.5\text{–}39.5\). From the intersection of the diagonals drawn in the modal rectangle on the histogram, the modal value is approximately \(33.5\).

Equivalently,

\[\text{Mode}=29.5+\frac{93-69}{(93-69)+(93-57)}\times 10\]

\[=29.5+\frac{24}{60}\times10=\boxed{33.5}\]

Thus, the estimated modal tyre life span is \(\boxed{33.5\times100=3350\text{ km}}\).

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Answer Details

(a) Histogram

Since all the class intervals have the same width, namely 10 (in units of 100 km), the bar heights are proportional to the frequencies. Use continuous class boundaries as follows:

Distance (100 km)	Class boundaries	Frequency
10–19	9.5–19.5	30
20–29	19.5–29.5	69
30–39	29.5–39.5	93
40–49	39.5–49.5	57
50–59	49.5–59.5	36
60–69	59.5–69.5	15

(b) Estimate of the mode

Equivalently,

\[\text{Mode}=29.5+\frac{93-69}{(93-69)+(93-57)}\times 10\]

\[=29.5+\frac{24}{60}\times10=\boxed{33.5}\]

Thus, the estimated modal tyre life span is \(\boxed{33.5\times100=3350\text{ km}}\).

Read lesson note on Statistics (WAEC)

Statistics

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A tyre manufacturing company researched into the life span of one type of their motorcycle tyres. The results were as follows : Distance (100km) 10-19 20-29...

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A tyre manufacturing company researched into the life span of one type of their motorcycle tyres. The results were as follows : Distance (100km) 10-19 20-29...

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