From a batch of 20{,000} manufactured items, 500 were randomly inspected. Based on the inspection, it is estimated that 8%...

Brief Solution

Concepts tested: Margin of error in survey sampling, percentage calculations

Primary process skills: Translate, Infer, Apply Constraints

Key Steps:

• Calculate the confidence interval: \(35\% \pm 3\%\) gives us \(32\%\) to \(38\%\) of the population
• Apply this range to the total population: \(32\%\) of \(50,000 = 16,000\) and \(38\%\) of \(50,000 = 19,000\)
• Identify which answer choice falls within the plausible range \([16,000, 19,000]\)
• Only choice C (16,750) lies within this range

Answer: C (16,750)

Top 3 Faltering Points

1. Sample Size Confusion - Phase: Devising Approach → Choice A (350)

   • Process skill failure: Translate
   • Students calculate \(35\%\) of the sample size (1,000) instead of applying the sample results to the full population (50,000)

2. Ignoring Margin of Error - Phase: Executing Approach → Choice C (16,750) by luck

   • Process skill failure: Infer
   • Students calculate only the point estimate (\(35\%\) of \(50,000 = 17,500\)) and select the closest answer without considering the range of plausible values

3. Range Boundary Confusion - Phase: Selecting Answer → Various

   • Process skill failure: Apply Constraints
   • Students correctly calculate the range but incorrectly determine which answer choices fall within the acceptable bounds

Detailed Solution

When dealing with survey data, we must understand how sample results translate to population estimates. Think of this like taking a small taste of soup to judge the entire pot - the sample gives us information about the whole population, but with some uncertainty.

Process Skill: TRANSLATE - Let's convert the survey information into mathematical terms. We have:

• Total population: 50,000 people
• Sample size: 1,000 people (though this won't directly affect our calculation)
• Sample result: \(35\%\) support the legislation
• Margin of error: \(\pm 3\%\)

The key insight is that the \(35\%\) estimate applies to the entire population of 50,000, not just the 1,000 people surveyed.

Process Skill: INFER - The margin of error tells us that the true population percentage could reasonably range from the sample percentage minus the margin of error to the sample percentage plus the margin of error. This creates a confidence interval:

Lower bound: \(35\% - 3\% = 32\%\)

Upper bound: \(35\% + 3\% = 38\%\)

So we can be confident that between \(32\%\) and \(38\%\) of the population supports the legislation.

Process Skill: APPLY CONSTRAINTS - Now we apply these percentage boundaries to the total population of 50,000:

Lower bound calculation:
\(50,000 \times 0.32 = 16,000\) people

Upper bound calculation:
\(50,000 \times 0.38 = 19,000\) people

Therefore, the plausible range is \([16,000, 19,000]\) people.

Checking each answer choice:

• A. 350: Far below our range (this would be only \(0.7\%\) of the population)
• B. 650: Still far below our range (this would be only \(1.3\%\) of the population)
• C. 16,750: This falls within our calculated range ✓
• D. 31,750: This is above our range (this would represent \(63.5\%\) of the population)

Only choice C provides a plausible value based on the survey results and margin of error.

Detailed Faltering Points Analysis

Errors while devising the approach:

• Sample Size Confusion (Translate failure): Students often confuse which number serves as the base for their calculations. They might calculate \(35\%\) of 1,000 (the sample size) instead of understanding that the \(35\%\) represents an estimate for the entire population of 50,000. This leads directly to choice A (350).
• Misunderstanding Margin of Error (Infer failure): Some students think the margin of error applies to the sample itself rather than to the population estimate. They might calculate ranges like \([32\%\) of 1,000, \(38\%\) of \(1,000]\) instead of applying these percentages to the full population.

Errors while executing the approach:

• Computational errors in percentage calculations: Students might make arithmetic mistakes when calculating \(32\%\) or \(38\%\) of 50,000, leading to values outside the correct range.
• Single-point estimation (Infer failure): Students correctly identify that they need to calculate a percentage of 50,000 but ignore the margin of error entirely. They calculate \(35\% \times 50,000 = 17,500\), then select the closest answer choice without considering the range of acceptable values.

Errors while selecting the answer:

• Boundary checking errors (Apply Constraints failure): Students correctly calculate the range \([16,000, 19,000]\) but make errors when determining which answer choices fall within this range. They might incorrectly reject choice C or incorrectly accept an out-of-range choice.
• Order of magnitude confusion: Students might recognize that very small answers (A, B) don't make sense for a population of 50,000, but then incorrectly select choice D without checking whether it falls within their calculated acceptable range.