AP Statistics 4 Inference for Quantitative Data Means Questions

Review AP Statistics Unit 4 through sampling distributions, t confidence intervals, t tests, matched pairs, and comparisons of means.

Exam points

describe sampling distributions of sample means and calculate standard errors and tail probabilities
construct and interpret one-sample or matched-pairs t-intervals after checking randomisation, independence and shape
state, perform and conclude one-sample or matched-pairs t-tests using ordered differences and contextual p-values
construct and interpret two-sample t-intervals for ordered mean differences using independent groups and unpooled standard errors
state, perform and conclude two-sample t-tests while distinguishing independent groups from paired data

Covered syllabus points

4.1 Sampling Distributions for Sample Means 4.2 Constructing a Confidence Interval for a Population Mean 4.3 Justifying a Claim Based on a Confidence Interval for a Population Mean 4.4 Setting Up a Test for a Population Mean or Population Mean Difference 4.5 Carrying Out a Test for a Population Mean or Population Mean Difference 4.6 Sampling Distributions for the Difference Between Two Population Means 4.7 Constructing a Confidence Interval for the Difference Between Two Population Means 4.8 Justifying a Claim Based on a Confidence Interval for the Difference Between Two Population Means 4.9 Setting Up a Test for the Difference Between Two Population Means 4.10 Carrying Out a Test for the Difference Between Two Population Means

[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]

A population is normally distributed with mean 25. Consider all samples of size 10 . The variable $\frac{\bar{x}-25}{\frac{s}{\sqrt{10}}}$

has a normal distribution.

has a t-distribution with d f=10.

has a t-distribution with d f=9.

has either a normal or a t-distribution depending on the characteristics of the population standard deviation.

[Maximum number: 1]

Recent studies have determined with $94 \%$ confidence that the mean number of trees cut down every 24 hours to make toilet paper is between 25,000 and 29,000. What is meant by the confidence level in this context?

A confidence interval of the true mean number of trees cut down every 24 hours to make toilet paper was calculated using z-scores of ±1.881.

A confidence interval of the true mean number of trees cut down every 24 hours to make toilet paper was calculated using t-scores consistent with d f=n-1 and tail probabilities of 0.03 .

We are $94 \%$ confident that the true mean number of trees cut down every 24 hours to make toilet paper is between 25,000 and 29,000.

If all possible random samples of data are obtained by this method, approximately 94\% will yield confidence intervals that capture the true mean number of trees cut down every 24 hours to make toilet paper.

[Maximum number: 8]

A top-100, 7.0-rated ten)nis pro wishes to compare a new racket against his current model. He is interested in whether his hitting speed with the new racket is different from that with his old racket. He strings the new racket with the same type of strings at 60 pounds tension that he uses on his old racket. From past testing, the tennis pro knows that the average forehand crosscourt volley with his old racket is 82 miles per hour (mph). On an indoor court, using a ball machine set at 70 mph, which is the same speed he had his old racket tested against, he takes 47 swings with the new racket. (Assume that swings are independent of each other.) An associate with a speed gun records an average of 83.5 mph with a standard deviation of 3.4 mph. At the 0.01 level of significance, complete the appropriate inference procedure to determine if there is convincing statistical evidence that his mean hitting speed with the new racket is different from that with his old racket.

(a)

Identify the appropriate inference procedure.

[ 1 ]

(b)

Complete the inference procedure, including calculating the appropriate statistics.

[ 5 ]

Let $\mu$ be the true mean hitting speed with the new racket. Test $H_0:\mu=82$ against $H_a:\mu\ne82$ . The swings are independent, and $n=47\ge30$ , so the normality condition is satisfied by the central limit theorem. The test statistic is t=3.0246, with df=46 and p=0.00406.

(c)

Justify a conclusion in context.

[ 2 ]

All question bank results loaded

Covered syllabus points

Question 1

Question (a)

Question (b)

Question (c)

Question 2

Question 3

Question 4

Question (a)

Question (b)

Question (c)