Sampling Distributions and the CLT - Tchebysheff’s Inequality - Central Limit Theorem - Statistics Assessment Answer

Assessment Task:

Sampling Distributions and the CLT Statistics Assessment Answer

TASK 1: The Central Limit Theorem.

You may wish to read the section 7.2, and 7.3 of Rice (2007). This task is designed to help you explore the central limit theorem (CLT). Repeat what follows for samples of size n ∈ {5,10,25,50,100,250,500} and number of iterations N ∈ {100,1 000}.

1. Generate n random numbers from an F-distribution with ν1 = 14 numerator and ν2 = 9 denominator degrees of freedom.

2. Determine the sample mean, y, of the sample.

3. Repeat the first two steps N times and store the values for y in a vector of length N.

4. Plot a relative frequency histogram for the sample means.

5. Add an appropriate density function of the sampling distribution.

6. Estimate the mean and the variance of the sampling distribution from your data and compare these to the theoretical values.

7. Comment on:

(a) The shape of the histograms.

(b) The mean and variance of your data compared to the theoretical values given by the Central Limit Theorem.

(c) The sample size, n, from when on you find the approximation given by the Central Limit Theorem ”good”. In addition plot the density of the F-distribution and a normal density with the mean and variance of the F distribution. Note that the density of the F distribution is very skewed and not even nearly normal.

TASK 2: Sampling from a Normal Distribution

This task is designed to help you explore the sampling distributions of Y and (n−1)S 2 σ2 when sampling from a normal distribution. In addition you will investigate the independence of Y and S 2 . To do this you will take repeated samples taken from a specific normal distribution. Relative frequency histograms will be used to visualize the sampling distributions of these statistics. The data will be used to estimate the parameters of these sampling distributions. You may wish to read the section 7.1, and 7.2 of Rice (2007).

You should repeat what follows for samples of size n ∈ {5,100} and number of iterations N ∈ {100,1 000,100 000} where µ = 5 and σ 2 = 2.

1. Generate n random numbers from a normal distribution with parameters µ and σ 2

2. Determine the sample mean, y, and the sample variance, s 2, of this particular sample.

3. Repeat the first two steps N times and store the values for y and s 2 in a data frame with N rows.

4. Plot a relative frequency histogram of the sample means, y.

5. Plot a relative frequency histogram of (n−1)S 2 σ2.

6. For each of the graphs in questions 4 and 5, add an appropriate density function of the sampling distribution. Clearly justify your choice of a sampling distribution.

7. Explore the independence of Y and S 2 : (a) Construct a scatter plot of the y’s the s 2 ’s; (b) Estimating the covariance of Y and S 2. From lectures we note that if Y and S 2 are independent then Covar Y, S 2  = 0; (c) Estimating the following probabilities: i. P 5 ≤ Y ≤ 5.3; ii. P 2 ≤ S 2 ≤ 2.5  ; iii. P 5 ≤ Y ≤ 5.3,2 ≤ S 2 ≤ 2.5

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