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AP Statistics 1.5 Representing a Quantitative Variable with Graphs- FRQs - Exam Style Questions

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Question

Corn tortillas are made at a large facility that produces \(100,000\) tortillas per day on each of its two production lines. The distribution of the diameters of the tortillas produced on production line A is approximately normal with mean \(5.9\) inches, and the distribution of the diameters of the tortillas produced on production line B is approximately normal with mean \(6.1\) inches. The figure below shows the distributions of diameters for the two production lines.
The tortillas produced at the factory are advertised as having a diameter of \(6\) inches. For the purpose of quality control, a sample of \(200\) tortillas is selected and the diameters are measured. From the sample of \(200\) tortillas, the manager of the facility wants to estimate the mean diameter, in inches, of the \(200,000\) tortillas produced on a given day. Two sampling methods have been proposed.
Method 1: Take a random sample of \(200\) tortillas from the \(200,000\) tortillas produced on a given day. Measure the diameter of each selected tortilla.
Method 2: Randomly select one of the two production lines on a given day. Take a random sample of \(200\) tortillas from the \(100,000\) tortillas produced by the selected production line. Measure the diameter of each selected tortilla.
(a) Will a sample obtained using Method 2 be representative of the population of all tortillas made that day, with respect to the diameters of the tortillas? Explain why or why not.
(b) The figure below is a histogram of \(200\) diameters obtained by using one of the two sampling methods described. Considering the shape of the histogram, explain which method, Method 1 or Method 2, was most likely used to obtain a such a sample.
(c) Which of the two sampling methods, Method 1 or Method 2, will result in less variability in the diameters of the \(200\) tortillas in the sample on a given day? Explain.
Each day, the distribution of the \(200,000\) tortillas made that day has mean diameter \(6\) inches with standard deviation \(0.11\) inch.
(d) For samples of size \(200\) taken from one day’s production, describe the sampling distribution of the sample mean diameter for samples that are obtained using Method 1.
(e) Suppose that one of the two sampling methods will be selected and used every day for one year (\(365\) days). The sample mean of the \(200\) diameters will be recorded each day. Which of the two methods will result in less variability in the distribution of the \(365\) sample means? Explain.
(f) A government inspector will visit the facility on June 22 to observe the sampling and to determine if the factory is in compliance with the advertised mean diameter of \(6\) inches. The manager knows that, with both sampling methods, the sample mean is an unbiased estimator of the population mean. However, the manager is unsure which method is more likely to produce a sample mean that is close to \(6\) inches on the day of sampling. Based on your previous answers, which of the two sampling methods, Method 1 or Method 2, is more likely to produce a sample mean close to \(6\) inches? Explain.

Most-appropriate topic codes (CED):

• TOPIC 3.2: Introduction to Planning a Study — part (a)
• TOPIC 1.5: Representing a Quantitative Variable with Graphs — part (b)
• TOPIC 3.3: Random Sampling and Data Collection — part (c)
• TOPIC 5.7: Sampling Distributions for Sample Means — parts (d), (e), (f)
▶️ Answer/Explanation
Detailed solution

(a)
No. Method 2 samples from only one production line. Since the two lines produce tortillas with different mean diameters (\(5.9\) vs \(6.1\) inches), a sample from just one line cannot represent the entire population from both lines.

(b)
Method 1 was likely used. The histogram is bimodal, suggesting the sample contains tortillas from both production lines (centered near \(5.9\) and \(6.1\) inches). Method 2 would likely produce a unimodal histogram.

(c)
Method 2 will result in less variability within a single sample. Method 2 samples from only one distribution (either Line A or Line B), while Method 1 samples from a mixture of two distributions with different centers, leading to greater spread in the combined sample data.

(d)
For Method 1 (\(n=200\), \(\mu=6\), \(\sigma=0.11\)):

  • Shape: Approximately normal (by CLT, since \(n=200 \ge 30\)).
  • Center: Mean \(\mu_{\bar{x}} = \mu = 6\) inches.
  • Spread: Standard deviation \(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{0.11}{\sqrt{200}} \approx 0.0078\) inches.

(e)
Method 1 will result in less variability in the \(365\) daily sample means.
Explanation: Sample means from Method 1 will cluster tightly around \(6\) inches (small \(\sigma_{\bar{x}}\)). Sample means from Method 2 will cluster around \(5.9\) inches about half the time and around \(6.1\) inches the other half, resulting in a much wider spread of the \(365\) sample means.

(f)
Method 1 is more likely to produce a sample mean close to \(6\) inches.
Explanation: Although both methods yield unbiased estimates in the long run, Method 1 has a sampling distribution with significantly less variability (as shown in (d) and (e)) and is centered exactly at \(6\). Therefore, any single sample mean from Method 1 is much more likely to be near \(6\) than a sample mean from Method 2, which will likely be near \(5.9\) or \(6.1\).

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