Estimation of Parameters - In Depth Notes

Estimation of Parameters - In Depth Notes

Point Estimates

Objectives

  • Find a point estimate for the population mean and the population proportion.

Key Concepts

  • Point Estimate: A single value that serves as an approximation of a population parameter. It provides a best guess based on sample data.

Central Limit Theorem (CLT)

  • Condition 1: If samples are drawn from any population with size ( n \geq 30 ), the sampling distribution of sample means will approximate a normal distribution.

  • Condition 2: If the population is normally distributed, the sampling distribution of sample means is also normally distributed for any size ( n ).

Example of Point Estimate

  • Suppose you want to estimate the average number of Facebook friends among senior high school (SHS) students in Quezon City.

    • Procedure: Gather a random sample of at least ( n = 30 ) Facebook users.

    • Sample Data: 140, 105, 130, 97, 80, 165, 232, 110, 214, 201…

    • Calculation: The sample mean (point estimate) is ( ar{x} = 132.7 ).

    • Conclusion: ( 132.7 ) is an unbiased point estimate of the population mean number of Facebook friends.

Other Examples of Point Estimators

Parameter

Statistic / Point Estimate

Mean

( c6 = ar{x} )

Variance

( c6^2 = s^2 )

Standard Deviation

( c6 = s )

Proportion

( p = \frac{x}{n} )

Interval Estimates

Objectives

  • Find an interval estimate for the population mean and population proportion.

Key Concepts

  • Interval Estimate: A range of values, derived from the sample statistics, likely to contain the true population parameter.

    • Point Estimate serves as the center of the interval.

    • Margin of Error (E) sets the range for the interval.

How to Form an Interval Estimate

  1. Point Estimate is the center of the interval.

  2. Calculate Margin of Error (E) to define the endpoints of the interval.

  3. Ensure ( n \geq 30 ) and use sample standard deviation if the population standard deviation is unknown.

Example of Confidence Interval

  • Sample Data: 140, 105, 130, 97, 80, 165…

  • Calculation: ( ar{x} = 132.7 ), sample standard deviation ( s = 54.2 )

  • For a 90% confidence interval:

    • Find ( z_c = 1.645 )

    • Margin of error ( E = z_c * \frac{s}{\sqrt{n}} )

    • Interpretation: With 90% confidence, the population mean number of Facebook friends is between 116.4 and 149.0.

Sample Size Computation

Objectives

  • Compute appropriate sample size for accurate estimates of parameters.

Key Concepts

  • Margin of Error (E): The maximum allowable difference between the sample statistic and the population parameter.

  • Standard Error: The standard deviation of the sampling distribution of the sample mean. It indicates accuracy of the sample mean as an estimate.

Minimum Sample Size Formula

  • If ( n \geq 30 ), use standard deviation ( s ).

  1. For population mean:
    [ n = \left( \frac{z_c * \sigma}{E} \right)^2 ]

  2. For population proportion:
    [ n = \left( \frac{z_c^2 * p(1-p)}{E^2} \right) ]

Example of Sample Size Calculation

  • Scenario: Students want to estimate daily solid waste weight within 5.2kg, assuming standard deviation is 17.6.

    • For 90% confidence, find ( n ):

    • The calculated sample size required is 31 days.

t-Distribution

Objectives

  • Find interval estimates for the population mean when ( n < 30 ) and ( \sigma ) is unknown.

Key Concepts

  • t-distribution: A probability distribution used when sample sizes are small ( ( n < 30 )) and population standard deviation is unknown.

    • Developed by William S. Gosset.

    • Characteristics include:

    • Bell-shaped and symmetric around the mean.

    • Varies by degrees of freedom (df); as df increases, it approaches normal distribution.

When to Use t-distribution

  • Use t-distribution when:

    • Sample size is less than 30.

    • Population standard deviation is not known.

Example of t-distribution

  • To find critical values for a 95% confidence level and sample size 15:

  1. Calculate degrees of freedom: ( df = n - 1 = 14 ).

  2. Reference t-table to find ( t_c ).

Summary of z-test vs t-test

  • z-test: Used for large sample sizes or known population standard deviation.

  • t-test: Used for small sample sizes or unknown population standard deviation.

Summary of Estimates

  • Point Estimates: Best single guesses of population parameters based on sample statistics.

  • Interval Estimates: Range likely containing the true population parameter, constructed using point estimates and margins of error.

  • Sample Size: Determining an adequate sample size is essential for reliable estimates, influenced by desired confidence levels and allowable error margins.

Formulas and Meaning of Symbols

  1. Point Estimate for Population Mean:

    • Formula: ( \bar{x} )

    • Meaning: Sample mean, point estimate of the population mean.

  2. Point Estimate for Population Variance:

    • Formula: ( s^2 )

    • Meaning: Sample variance, point estimate of the population variance.

  3. Point Estimate for Population Standard Deviation:

    • Formula: ( s )

    • Meaning: Sample standard deviation, point estimate of the population standard deviation.

  4. Point Estimate for Population Proportion:

    • Formula: ( p = \frac{x}{n} )

    • Meaning: Proportion of successes in the sample.

  5. Sample Size for Population Mean:

    • Formula:
      [ n = \left( \frac{z_c \cdot \sigma}{E} \right)^2 ]

    • Meaning: Formula used to compute the necessary sample size for estimating the population mean.

  6. Sample Size for Population Proportion:

    • Formula:
      [ n = \left( \frac{z_c^2 \cdot p(1-p)}{E^2} \right) ]

    • Meaning: Formula to compute the necessary sample size for estimating population proportion.

  7. Margin of Error:

    • Formula:

    • Formula: ( E = z_c \cdot \frac{s}{\sqrt{n}} )

    • Meaning: The maximum allowable difference between the sample statistic and the population parameter.

  8. Confidence Interval Calculation:

    • Formula:
      [ CI = ( \bar{x} - E, \bar{x} + E ) ]

    • Meaning: The range likely containing the true population mean, where ( E ) is the margin of error.

  9. t-distribution critical value for confidence level:

    • Formula: Based on degrees of freedom (df):
      [ df = n - 1 ]

    • Meaning: Necessary to find critical values for small sample sizes or unknown population standard deviation.

Meanings of Symbols

  • ( \bar{x} ): Sample mean (point estimate of population mean)

  • ( s ): Sample standard deviation (point estimate of population standard deviation)

  • ( s^2 ): Sample variance (point estimate of population variance)

  • ( z_c ): z-value corresponding to desired confidence level

  • ( n ): Sample size

  • ( E ): Margin of error

  • ( \sigma ): Population standard deviation (if known)

  • ( p ): Proportion of successes in the sample

  • ( x ): Number of successes in the sample

  • ( CI ): Confidence interval

  • ( df ): Degrees of freedom (n - 1)

This summarizes all the formulas and their respective meanings for the symbols used in the context of parameter estimation.