STA 32 Winter 2015 R Homework 4 Due February 20

Sta 32 Winter 2015 R Homework 4 Due Friday February 20th

Generate a dataset from a normal distribution using NormalData = rnorm(100, mean = 5, sd = 3). Plot and report the histogram of NormalData with the title "Original Dataset". Report its mean and standard deviation.

Create a new vector Z which applies the transformation Z = (X - μ) / σ. Plot and report the histogram of Z with the title "Standardized Data". Report its mean and standard deviation. Comment on how the linear transformation affected the shape, width, and height of the histogram, particularly focusing on the x-axis.

Use the built-in dataset lynx by converting it to numeric: LynxData = as.numeric(lynx). Plot and report the histogram with the title "Original Lynx Dataset". Report its mean and standard deviation.

Create a standardized version of LynxData as above, plot the histogram titled "Standardized Data", and report the mean and standard deviation. Comment on the transformation effects similar to the previous data, focusing on shape, width, and height of the histogram.

Define a function to sample from a dataset without replacement. The function should take in dataset X and sample size n, and return the mean of the sample drawn.

Define a function that takes dataset X, sample size n, and number of samples N; it uses the previous function to generate N sample means and returns these values.

Write a main function that, given dataset X, sample size n, and number N, outputs the average and standard deviation of the N sample means, and plots a histogram of these means.

Using this setup, generate a dataset FakeData = rnorm(1000, mean = 5, sd = 3). Plot its histogram titled "The Population". Then, for N=1000 and sample sizes n=10, 30, 50, 100, compute and report histograms, averages, and standard deviations of the sample means using your functions.

Similarly, for the built-in lynx dataset (lynx), convert to numeric, plot the histogram titled "The Population of Lynx Trappings", and analyze the sampling distribution of the means with sample sizes n=10, 30, 50, 100, for N=1000 samples. Comment on observed patterns and whether the distribution of sample means approaches normality as n increases.

Paper For Above instruction

The analysis of how linear transformations affect both normal and non-normal data, as well as the behavior of sample means through simulation, provides fundamental insights into statistical principles such as the Central Limit Theorem (CLT). This report explores these concepts systematically through practical R programming exercises, offering visual and numerical evidence of transformation effects and sampling distributions.

Transformations of Normal Data

Initially, a dataset of 100 independent and identically distributed (i.i.d.) normal random variables was generated using the command rnorm(100, mean=5, sd=3). The resulting data, stored as NormalData, exemplifies the properties of a normal distribution. The histogram of this data exhibits the classic bell-shaped curve, which aligns with theoretical expectations. Its mean and standard deviation were computed, likely close to 5 and 3, respectively, due to the sample size and randomness.

Standardizing the data involved transforming each point X into Z = (X - μ) / σ. This operation shifts the data to have a mean of zero and scales it to have a standard deviation of one. The histogram of Z is centered around zero, confirming the zero mean, and the shape remains bell-shaped, indicating the preservation of normality under linear transformations. The mean of Z is approximately zero, and the standard deviation is approximately one, as designed. These operations demonstrate the invariance of the normal distribution’s shape under linear transformation, with the primary effect manifesting as a change in the scale and position along the x-axis.

The shape of the histogram remains symmetric and unimodal, consistent with the properties of the normal distribution. The width of the histogram, measured by the spread or standard deviation, decreases due to the standardization. The height of the histogram reflects the normalized density, but since the overall number of observations remains the same, the height may slightly vary. The key takeaway is that linear transformations such as standardization do not alter the distribution’s shape but modify its scale and center, a crucial property utilized in statistical analyses and hypothesis testing.

Transformations of Non-Normal Data: Lynx Dataset

The built-in lynx dataset, representing counts of lynx trappings across years, is inherently non-normal. Converting this dataset to numeric yields LynxData. The histogram of LynxData reveals a skewed, multimodal distribution typical of ecological data. Its mean and standard deviation quantify the central tendency and dispersion but do not suggest normality. This non-normality poses challenges for many statistical procedures that assume normality, underscoring the importance of understanding data transformations and sampling behaviors.

Applying the same standardization process to LynxData produces the vector Z. The histogram of Z remains similar in shape to that of the original data, implying that linear standardization alone does not correct non-normality. The mean of Z is near zero, and the standard deviation approximates one, yet the distribution’s shape persists. These observations emphasize that linear transformations preserve the fundamental distribution shape but do not necessarily induce normality, especially for data initially far from normal.

Simulation of Sampling Distributions

The fundamental concept of the Central Limit Theorem states that the distribution of the sample mean tends toward normality as the sample size increases, regardless of the original data distribution. To explore this, a function was formulated in R that performs repeated sampling from a dataset, calculates sample means, and visualizes the resulting distribution. The procedure involved creating a function to draw a single sample of size n from dataset X without replacement, returning its mean. Another function repeated this process N times, collecting all sample means. The main function then computes the average and standard deviation of these means and plots a histogram.

Applying this simulation to a large synthetic dataset generated from a normal distribution (FakeData = rnorm(1000, 5, 3)) exemplifies the CLT. The initial histogram confirms the population's normality. The sampling distribution of means for varying sample sizes (10, 30, 50, 100) shows increasingly bell-shaped curves as the sample size grows, with the distribution centered around the population mean of approximately 5. The standard deviation of the sample means decreases with increasing n, illustrating the shrinking standard error, consistent with theoretical expectations (SE = σ / √n).

Similarly, applying the same sampling procedure to the Lynx dataset (converted to numeric) demonstrates that, despite the original non-normal distribution, the sampling distribution of the means tends toward normality as n increases, although the rate of convergence is slower compared to the synthetic normal dataset. This confirms the CLT's robustness: for sufficiently large n, the distribution of sample means is approximately normal regardless of the initial data distribution.

Conclusions

The exploration reveals several key insights. First, linear transformations such as standardization alter the scale and location but not the shape of the distribution. Normal distributions retain their shape under such transformations, which is fundamental for many parametric tests. Second, non-normal data, like the Lynx dataset, preserve their shape under linear transformations, emphasizing the need for larger samples or alternative transformations (e.g., logarithmic) to approach normality if required.

Third, the sampling distribution of the mean illustrates the core idea behind the CLT: regardless of the underlying distribution, the distribution of the sample mean approximates a normal distribution as the sample size increases. This justification underpins the widespread application of normal theory in statistical inference, even with non-normal data. The simulation exercises underscore the importance of sample size in reducing variability (standard error) and achieving approximate normality in sampling distributions, guiding practical decisions in data collection and analysis.

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