DAT 301 Homework 3

Author: Pruthvi J Arizona State Univerity DATE: October 18,2025

The point estimation is a statistical technique that approximates an unknown population parameter, which is calculated using a single value (the point estimate) based on sample data.

Examples of point estimates: - Sample mean (\(\bar{x}\)) is a point estimate of population mean (\(\mu\)) -Sample Variance (\(s^2\)) is a point estimate of population variance (\(\sigma^2\))

Point estimate is fundamental in statistical analysis and inference

• Sample Mean (\(\bar{x}\)):Estimates the population mean (\(\mu\))
• Sample Proportion (\(\hat{p}\)): Estimates the population proportion (\(p\))
• Sample Variance (\(s^2\)): Estimates the population variance (\(\sigma^2\))

Example Values: - For sample data: 2,5,6, - Sample mean: \(\bar{x}=\frac{2+5+6}{3}=4.33\) - Sample Variance: \(\frac{(2-4.33)^2+(5-4.33)^2+(6-4.33)^2}{3-1}=4.33\)

Desirable Properties of good estimators: - Unbiasedness: The expected value of the estimator equals the true parameter value. - Consistency: The estimator approaches the true parameter as the sample size increases. - Efficiency: The estimator has the smallest possible vairance among all unbiased estimators.

Example: The sample mean \(\bar{x}\) is an unbiased estimator for the population mean \(\mu\): \[ E[\bar{x}]=\mu \]

Suppose we collect a random sample of size 10 from a normal distribution with unkown mean.

We can use the sample mean as a point estimator for the population mean.

## [1] 11.09459

The sample mean calculated above is our point estimate for the population mean.

The sample mean provides a point estimate of the population mean. Let’s visualize our sample data and the sample mean using a histogram.

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Let’s compare the sample mean and sample median as point estimators using a line plot.

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We can use a 3D scatter plot to visualize the relation between simulated sample means,sample sizes and the true mean.

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Some important formulas in point estimation:

• Sample mean estimator for the population mean: \[ \bar{x}=\frac{1}{n} \sum_{i=1}^n x_i \]

• Sample proportion estimator for population proportion: \[ \hat{p}=\frac{x}{n} \]

• Unbiasedness property of the sample mean: \[ E[\bar{x}]=\mu \]

• Consistency:

  An estimator \(\hat{\theta}_n\) is consistent for parameter \(\theta\) if: \[ \lim_{n \to \infty} P(|\hat{\theta}_n - \theta| <\epsilon)=1, \quad \text{for all } \epsilon >0 \]