Chapter 5 of your AP Statistics curriculum likely covers sampling distributions—a crucial concept for understanding inferential statistics. This practice test will help you solidify your understanding and prepare for the AP exam. Remember, mastering this chapter is key to success in later units.
Understanding Sampling Distributions: Key Concepts
Before we dive into the practice problems, let's review the core ideas of Chapter 5:
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Parameter vs. Statistic: A parameter is a numerical characteristic of a population, while a statistic is a numerical characteristic of a sample. Understanding this distinction is fundamental.
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Sampling Distribution: The sampling distribution of a statistic is the probability distribution of that statistic based on all possible samples of a given size from the population. It describes how the statistic varies from sample to sample.
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Central Limit Theorem (CLT): This is arguably the most important concept in Chapter 5. The CLT states that the sampling distribution of the sample mean (x̄) will be approximately normal, regardless of the shape of the population distribution, as long as the sample size (n) is sufficiently large (generally, n ≥ 30). The mean of this sampling distribution is equal to the population mean (μ), and the standard deviation is σ/√n (the standard error).
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Standard Error: The standard deviation of the sampling distribution of a statistic is called its standard error. It quantifies the variability of the statistic from sample to sample. A smaller standard error indicates less variability and more precision in estimating the population parameter.
AP Stats Chapter 5 Practice Test Questions
Here are some practice questions covering key aspects of Chapter 5. Try to answer them without looking at your notes, then check your work against the solutions provided below.
Question 1: Explain the difference between a population mean (μ) and a sample mean (x̄). Why is the sampling distribution of x̄ important?
Question 2: A population has a mean of 70 and a standard deviation of 10. If you take a random sample of size 49, what is the mean and standard deviation of the sampling distribution of the sample mean (x̄)?
Question 3: The distribution of heights of adult women is approximately normal with a mean of 65 inches and a standard deviation of 3 inches. If we take random samples of size 16, what is the probability that the sample mean height will be less than 64 inches? (Assume the CLT applies.)
Question 4: Explain why the Central Limit Theorem is so important in statistical inference.
Question 5: A researcher is studying the average time students spend studying for their AP Statistics exam. They take a random sample of 50 students and find that the average study time is 8 hours with a standard deviation of 2 hours. Construct a 95% confidence interval for the population mean study time. (Assume the CLT applies and the population standard deviation is unknown)
Solutions and Explanations
Question 1: A population mean (μ) is the average of all values in the entire population. A sample mean (x̄) is the average of the values in a subset of the population (the sample). The sampling distribution of x̄ is important because it allows us to make inferences about the population mean (μ) based on sample data. It tells us how much the sample mean is likely to vary from sample to sample.
Question 2: According to the Central Limit Theorem, the mean of the sampling distribution of x̄ is equal to the population mean (μ = 70). The standard deviation of the sampling distribution (the standard error) is σ/√n = 10/√49 = 10/7 ≈ 1.43.
Question 3: First, we need to find the standard error: SE = σ/√n = 3/√16 = 0.75. Then, we standardize the sample mean: z = (64 - 65) / 0.75 ≈ -1.33. Using a z-table or calculator, we find the probability that z < -1.33 is approximately 0.0918.
Question 4: The Central Limit Theorem is vital because it allows us to use the normal distribution to make inferences about population means, even when the population distribution is not normal, provided the sample size is large enough. This greatly simplifies statistical inference.
Question 5: Since the population standard deviation is unknown, we use the t-distribution. With a sample size of 50, the degrees of freedom are 49. For a 95% confidence interval, the t-critical value (using a t-table or calculator) is approximately 2.01. The margin of error is t* (s/√n) = 2.01 * (2/√50) ≈ 0.57. The 95% confidence interval is 8 ± 0.57, or (7.43, 8.57) hours.
This practice test provides a good starting point. Remember to review your textbook, class notes, and additional practice problems for a more comprehensive understanding of sampling distributions. Good luck with your AP Statistics exam!
