Mathematics
Grade 12
15 min
Distributions of sample means
Distributions of sample means
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1
Introduction & Learning Objectives
Learning Objectives
Model a probability density function for a distribution of sample means using a rational function.
Verify that a given rational function is a valid probability density function by showing its integral over the real numbers is 1.
Calculate the expected value (mean) of a sample mean distribution described by a rational function using definite integrals.
Use the first derivative to find the mode (peak value) of a distribution modeled by a rational function.
Interpret the horizontal asymptotes of a rational function in the context of the tail behavior of a probability distribution.
Explain the relationship between the parameters of a rational function and the characteristics (center, spread) of the distribution it models.
How can a company confidently claim t...
2
Key Concepts & Vocabulary
TermDefinitionExample
Sample Mean (x̄)The average of a set of data points taken from a larger population. It is a statistic used to estimate the population mean.If we test the battery life of 5 phones and get {18, 20, 21, 19, 22} hours, the sample mean x̄ is (18+20+21+19+22)/5 = 20 hours.
Distribution of Sample MeansThe probability distribution of all possible sample means (x̄) that could be obtained from a population for a given sample size (n).If we repeatedly took samples of 5 phones and calculated the mean battery life for each sample, the collection of all those means would form its own distribution, which tends to be bell-shaped.
Probability Density Function (PDF)A function, f(x), used to describe the probability of a continuous random variable. The area under the curve of a PDF ove...
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Core Formulas
PDF Normalization Condition
∫_{-∞}^{∞} f(x) dx = 1
For any function f(x) to be a valid probability density function, the total area under its curve across its entire domain must be exactly 1. This represents 100% of the total probability.
Expected Value (Mean) of a Continuous Distribution
μ = E[X] = ∫_{-∞}^{∞} x * f(x) dx
This formula calculates the mean (μ) or expected value (E[X]) of a continuous random variable X with a PDF of f(x). It is a weighted average where each value x is weighted by its probability density f(x).
Central Limit Theorem (Core Idea)
μ_{x̄} = μ and σ_{x̄} = σ / √n
While we model the shape with rational functions, the Central Limit Theorem provides the theoretical center and spread for the distribution of sample means. The mean of the sample m...
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Challenging
The distribution of sample means for an experiment with sample size n=20 is well-modeled by f(x) = (2/π) / (4 + x²). According to the Central Limit Theorem, if the sample size is increased to n=80, the distribution of sample means will have a smaller variance (be less spread out). Which function could model the new distribution?
A.g(x) = (3/π) / (9 + x²)
B.g(x) = (1/π) / (1 + x²)
C.g(x) = (2/π) / (4 + (x-2)²)
D.g(x) = 4 * (2/π) / (4 + x²)
Challenging
For a distribution of sample means modeled by a rational function of the form f(x) = k / (a² + (x-b)²), where a > 0, what is the general expression for the normalization constant k in terms of the parameter a?
A.k = a/π
B.k = π/a
C.k = a²/π
D.k = 1/π
Challenging
For a distribution of sample means with the PDF f(x) = (1/π) / (1 + x²), calculate the expected value of X², denoted E[X²]. Note: This integral is improper and does not converge.
A.0
B.1
C.1/2
D.Undefined (the integral diverges)
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