AP Statistics Unit 4: Inference for Means
Review AP Statistics Unit 4 through sampling distributions, t confidence intervals, t tests, matched pairs, and comparisons of means.
- Syllabus
- Effective Fall 2025
- Course
- AP Statistics
Covered syllabus points
4.1 Sampling Distributions for Sample Means4.2 Constructing a Confidence Interval for a Population Mean4.3 Justifying a Claim Based on a Confidence Interval for a Population Mean4.4 Setting Up a Test for a Population Mean or Population Mean Difference4.5 Carrying Out a Test for a Population Mean or Population Mean Difference4.6 Sampling Distributions for the Difference Between Two Population Means4.7 Constructing a Confidence Interval for the Difference Between Two Population Means4.8 Justifying a Claim Based on a Confidence Interval for the Difference Between Two Population Means4.9 Setting Up a Test for the Difference Between Two Population Means4.10 Carrying Out a Test for the Difference Between Two Population Means
4 Inference for Quantitative Data: Means question 1
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.
Question (a)
(a)
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.
Question (b)
(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.
Question (c)
(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.
4 Inference for Quantitative Data: Means question 2
A population is normally distributed with mean 25. Consider all samples of size 10 . The variable
A
has a normal distribution.
B
has a t-distribution with d f=10.
C
has a t-distribution with d f=9.
D
has either a normal or a t-distribution depending on the characteristics of the population standard deviation.
4 Inference for Quantitative Data: Means question 3
Recent studies have determined with 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
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.
B
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 .
C
We are confident that the true mean number of trees cut down every 24 hours to make toilet paper is between 25,000 and 29,000.
D
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.
4 Inference for Quantitative Data: Means question 4
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.
Question (a)
(a)
Identify the appropriate inference procedure.
[ 1 ]
Question (b)
(b)
Complete the inference procedure, including calculating the appropriate statistics.
[ 5 ]
Question (c)
(c)
Justify a conclusion in context.
[ 2 ]
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