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

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Required skills (click to reveal)

A craft designer in Brisbane creates two batches of ceramic tiles. Batch A contains 38 tiles with a mean weight of 720 grams and a standard deviation of 14 grams.

Batch B contains 48 tiles with a mean weight of 720 grams and a standard deviation of 21 grams. The designer selects a random sample of 6 tiles from Batch A and 6 tiles from Batch B.

What is the exact difference between the sample means of the two batches, given that the samples are drawn from populations with identical means? Express your answer as an exact integer or 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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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The question states that both Batch A and Batch B have the same population mean weight of 720 grams. The sample means are drawn from these populations, and the question asks for the exact difference between the sample means of two random samples of 6 tiles each.
  2. Since the samples are drawn from populations with identical means (720 grams), the expected value of the sample mean for each batch is 720 grams. Therefore, the expected difference between the two sample means is 720 - 720 = 0 grams.
  3. The standard deviation of Batch A is 14 grams, and Batch B is 21 grams. However, the question does not ask for the standard error of the difference or any measure of variability. It only asks for the exact difference between the sample means, given that the population means are identical.
  4. The actual sample means will vary due to sampling variability, but the question specifies that the samples are drawn from populations with identical means. The expected difference between the sample means is zero, and since no specific sample values are given, the only exact answer consistent with the information is zero.
  5. The question asks for the exact difference between the sample means, not a probability or a confidence interval. Since no actual sample data is provided, and the population means are equal, the only exact value that satisfies the condition is 0 grams.
  6. No further calculation is needed because the expected difference between two sample means drawn from populations with identical means is exactly zero, regardless of sample size or standard deviation.
  7. The standard deviation of the samples (14 g and 21 g) affects the variability of the sample means but not their expected difference. Therefore, these values are not used in computing the exact difference.
  8. The sample size (6 tiles) is the same for both batches, but this does not change the expected difference between the means, which remains zero.
  9. The question does not provide actual sample weights, so no numerical difference can be computed from data. The only exact answer based on the given information is 0 grams.
  10. The difference between the sample means is a random variable, but the question asks for the exact difference under the given condition. Since the population means are equal, the expected difference is exactly zero.
  11. Therefore, the exact difference between the sample means is 0 grams.
  12. This result is consistent with the principle that the expected difference between two sample means from populations with identical means is zero, regardless of sample size or standard deviation.

Reference answer: 0 grams

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