AP Statistics 4.1 Sampling Distributions

[Maximum number: 4]

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.

Scoring

This question is scored in three sections. Section 1 consists of parts (a), (b), and (c), section 2 consists of part (d), and section 3 consists of parts (e) and (f). Sections 1, 2, and 3 are scored as essentially correct (E), partially correct (P), or incorrect (I).

Section 1 is scored as follows:
Essentially correct (E) if the response includes the following three components:
1. In part (a) the response says no A N D either argues that the sample will only be selected from one production line and not the entire population (that is, the sample will only represent one production line) O R argues that the tortillas from the two productions lines are different.
2. In part (b) the response chooses Method 1 AND refers to a relevant characteristic of the histogram (shape, center, or variability) that matches what would be expected when using Method 1 (or that does not match what would be expected when using Method 2).
3. In part (c) the response chooses Method 2 AND either justifies by stating that the sample comes from only one production line (does not come from both production lines) OR justifies by comparing the possible range of diameters for the two methods.

Partially correct (P) if the response includes only two of the three components.
Incorrect (I) if the response includes at most one of the three components.
Note: If a response includes more than one justification in an individual part, score the weaker of the two justifications. For example, a response for part (a) that says "No, because only one line was selected and because the sample size was too small" does not satisfy the first component because the sample size argument is incorrect.

Section 2 is scored as follows:
Essentially correct (E) if the response includes the following three components:
1. The shape is approximately normal.
2. The mean is 6 inches.
3. The standard deviation is $\frac{0.11}{\sqrt{200}}$ inch.

Partially correct (P) if the response includes only two of the three components.
Incorrect (I) if the response includes at most one of the three components.
Note: It is not necessary to include units (inches) for the mean or standard deviation.
Section 3 is scored as follows:
Essentially correct (E) if the response includes the following two components:
1. In part (e) the response chooses Method 1 AND describes the sampling distribution of the sample mean for Method 2 as having some sample means close to the mean of production line A (5.9 inches) and the other sample means close to the mean of production line B (6.1 inches).
2. In part (f) the response chooses Method 1 AND
- refers to a correct answer in part (e);
OR
- describes the sampling distribution of the sample mean for Method 2 as having some sample means close to the mean of production line A (5.9, less than 6) and the other sample means close to the mean of production line B (6.1, greater than 6);
OR
- argues that on a single day, it would be preferable to get a sample with a mean around 6 rather than getting a sample that would have a mean around 5.9 (less than 6) or a mean around 6.1 (greater than 6).

Partially correct (P) if the response includes one of the two components.
Incorrect (I) if the response does not meet the criteria for E or P.
Notes:
- In part (e), the response must be clear that there is more than one mean being described (365 sample means).
- Correct: The sample means will be around 5.9 or 6.1.
○ Not correct: The sample mean will be around 5.9 or 6.1.
- Parts (e) and (f) are not satisfied if the response does not imply the sample means vary from the population means in both methods (or does not imply that the sample mean varies from the population mean when making the single day argument in part (f)). However, if the mistake is made in part (e), do not penalize the response in part (f) for the same mistake.
- Correct: The sample means will be around 5.9 or 6.1 instead of close to 6 .
- Not correct: The sample means will be 5.9 or 6.1 instead of close to 6.
- Not correct: The sample means will be 6 instead of around 5.9 or 6.1.
4 Complete Response
All three sections essentially correct
3 Substantial Response
Two sections essentially correct and one section partially correct
2 Developing Response
Two sections essentially correct and one section incorrect
OR
One section essentially correct and either one or two sections partially correct
OR
All three sections partially correct
1 Minimal Response
One section essentially correct and two sections incorrect
OR
Two sections partially correct and one section incorrect

(a)

Each day, the distribution of the 200,000 tortillas made that day has mean diameter 6 inches with standard deviation 0.11 inch.

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.

Part (d):

The sampling distribution of the sample mean diameter for samples obtained using Method 1 would be approximately normal with mean 6 inches and standard deviation $\frac{0.11}{\sqrt{200}} \approx 0.0078$ inch.

(b)

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.

(c)

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.

[Maximum number: 1]

The sampling distribution of the sample mean is close to the normal distribution

only if both the original population has a normal distribution and n is large.

if the standard deviation of the original population is known.

if n is large, no matter what the distribution of the original population.

only if the original population is not badly skewed and does not have outliers.

AP Statistics 4.1 Sampling Distributions for Sample Means Questions

Exam points

Covered syllabus points

Covered syllabus points

Question 1

Question (a)

Question (b)

Question (c)

Question 2