NavGanitMateby NavGreen Compute, AustraliaBased on Victorian Curriculum Mathematics v2.0
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Comparing data distributions

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A community garden in Brisbane has two vegetable plots, each with 63 plants. Plot A has a mean yield of 87 grams per plant with a standard deviation of 12 grams.

Plot B has a mean yield of 87 grams per plant with a standard deviation of 18 grams. A random sample of 7 plants is taken from each plot.

What is the exact difference between the sample means of Plot A and Plot B, given that both samples are drawn from populations with identical means? Express your answer as an exact fraction.

Concept illustrations

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A five-number summary. Illustrative five-number summary: minimum 1, Q1 4, median 7, Q3 10, maximum 13. IQR = 10 − 4 = 6. Whiskers use the minimum/maximum convention.1471013IQR = Q3 − Q1 = 6
Illustrative five-number summary: minimum 1, Q1 4, median 7, Q3 10, maximum 13. IQR = 10 − 4 = 6. Whiskers use the minimum/maximum convention.
Median, mode and range in sorted data. The ordered values are 2,3,3,8,9. The middle value is 3, the most frequent value is 3, and the range is 9 − 2 = 7. The row uses equal spacing, not a number-line scale.23389↑ medianmode 3 range 7
The ordered values are 2,3,3,8,9. The middle value is 3, the most frequent value is 3, and the range is 9 − 2 = 7. The row uses equal spacing, not a number-line scale.

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Formula

Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The question states that both Plot A and Plot B have identical population means of 87 grams per plant. A random sample of 7 plants is taken from each plot.
  2. The sample mean is a random variable that, for a given sample size, has a distribution centered on the population mean. Since both populations have the same mean, the expected value of the sample mean for both plots is 87 grams.
  3. The question asks for the exact difference between the sample means of Plot A and Plot B, given that both samples are drawn from populations with identical means.
  4. Because the population means are identical and the samples are drawn independently, the expected difference between the two sample means is zero. This is a property of the sampling distribution of the difference between two independent sample means.
  5. The standard deviations of the plots (12 g and 18 g) and the sample size (7) are given, but they affect the variability of the sample means, not their expected difference.
  6. The question does not ask for a probability, a confidence interval, or a measure of variability. It asks for the exact difference between the sample means, given that the population means are equal.
  7. Since the population means are equal and the samples are random and independent, the expected value of the difference between the two sample means is 0 grams.
  8. The actual difference in any given sample will vary due to sampling variability, but the question asks for the exact difference under the given condition of identical population means.
  9. The expected difference is not a random outcome but a theoretical value derived from the assumption of equal population means.
  10. Therefore, the exact difference between the sample means, under the given condition, is 0 grams.
  11. This result is exact and does not depend on the standard deviations or sample size, as the question specifies identical population means.
  12. The final answer is 0 grams, expressed as the exact fraction 0/1.

Reference answer: 0

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