Statistical Estimation Overview

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  1. Introduction
    1. Point Estimation
    2. Interval Estimation </aside>

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  1. Interval Estimation of Population Mean
    1. When Population Standard Deviation, Sigma(σ), is Known
    2. When Population Standard Deviation, Sigma(σ), is Unknown
    3. Sample Size Determination </aside>

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  1. Interval Estimation of Population Proportion
    1. Interval Estimation of Population Proportion
    2. Sample Size Determination </aside>

Introduction

  1. Statistical Estimation: Point and Interval Estimation

    1. Statistical estimation is a method used to infer information about a population based on data from a sample. It involves two main approaches: point estimation and interval estimation.

  2. Point Estimation

    1. Definition
      1. Point Estimation involves estimating the value of a population parameter based on a single value, called the point estimate, obtained from a sample statistic.
    2. Key Terms

      1. Point Estimator: The sample statistic used to estimate the corresponding population parameter.

      2. Point Estimate: The specific numerical value of the sample statistic.

      3. Population Parameter: The true value of the characteristic in the population.

      4. Correspondence Between Sample Statistic and Population Parameter

        Measure of Characteristic Sample Statistic Population Parameter
        Mean $\bar{x}$ $\mu$
        Standard Deviation $s$ $\sigma$
        Proportion of Success $\hat{p}$ $p$

      5. Example

        1. A sample of 50 students is taken, and their average test score ($\bar{x}$) is calculated as 78. Here:

          $\bar{x} = 78$ is the point estimate of the population mean $\mu$.

  3. Interval Estimation

    1. Definition

      1. Interval Estimation involves estimating the range (interval) within which the true value of a population parameter is likely to lie, with a certain level of confidence.

    2. Formula

      $$ \text{Interval Estimate} = \text{Point Estimate} \pm \text{Margin of Error} $$

      Where:

      1. Point Estimate: The sample statistic (e.g., $\bar{x}$ or $\hat{p}$).
      2. Margin of Error: A measure that reflects the level of uncertainty or variability in the estimate.

    3. Interval Estimates for Common Parameters

      1. Population Mean:

        $$ \text{Interval Estimate of Mean} = \bar{x} \pm \text{Margin of Error} $$

      2. Population Proportion:

        $$ \text{Interval Estimate of Proportion} = \hat{p} \pm \text{Margin of Error} $$

      3. Confidence Level Table:

        Confidence Level Alpha Alpha/2 z alpha/2
        90% 10% 5.0% 1.645
        95% 5% 2.5% 1.96
        98% 2% 1.0% 2.326
        99% 1% 0.5% 2.576

  4. Problem 1: Interval Estimate for Population Mean

    1. Given:

      1. Sample mean ($\bar{x}$) = 50
      2. Standard deviation ($\sigma$) = 5
      3. Sample size ($n$) = 25
      4. Confidence level = 95%

    2. Step 1: Identify the Formula

      $$ \text{Margin of Error} = z \times \frac{\sigma}{\sqrt{n}} $$

      Where:

      1. $z$ is the z-score corresponding to the confidence level (for 95%, $z = 1.96$),
      2. $\sigma$ is the population standard deviation,
      3. $n$ is the sample size.

    3. Step 2: Calculate the Standard Error

    4. Step 3: Calculate the Margin of Error

    5. Step 4: Compute the Interval Estimate

    6. Result: The 95% confidence interval for the population mean is $(48.04, 51.96)$.

  5. Problem 2: Interval Estimate for Population Proportion