A farmer inspected every field on his farm at harvest and classed the yield as "Good", "Average" or "Poor". Of the fields, 15 were classed "Good" and 9 were...
Question 1 Report
A farmer inspected every field on his farm at harvest and classed the yield as "Good", "Average" or "Poor". Of the fields, 15 were classed "Good" and 9 were classed "Average"; the rest were classed "Poor".
| Yield | Good | Average | Poor |
|---|---|---|---|
| Frequency | 15 | 9 | ? |
The relative frequency of a field being classed "Poor" was 0.2.
- Work out the total number of fields on the farm. (2)
- Write down the number of fields classed "Poor". (1)
- Give the probability that a field chosen at random is not "Poor", writing your answer as a fraction in its simplest form. (2)
Answer Details
As with any unknown-total relative-frequency problem, the "Poor" frequency can be expressed two ways, once as whatever is left after the known categories, and once as a proportion of the total, and the two expressions are set equal to find the total.
- (a) [2]. The known categories sum to \(15+9=24\), so the "Poor" frequency is \(T-24\), where \(T\) is the total. Since the relative frequency of "Poor" is \(0.2\), the "Poor" frequency is also \(0.2T\). So: \[T - 24 = 0.2T \implies 0.8T = 24 \implies T = 30 \text{ fields}\]
- (b) [1]. Fields classed "Poor" number \(30 - 24 = 6\).
- (c) [2]. "Not Poor" fields number \(24\), so \(P(\text{not Poor}) = \dfrac{24}{30} = \dfrac{4}{5}\).
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