7.1-7.3

7.2 Estimation and Test for a Population Variance

Introduction to Population Variance

  • The focus is on estimating the population variance (𝜎²) and testing claims regarding this variance.

Importance of Variance in Manufacturing

  • In manufacturing, it's crucial to monitor the variation of measurements, such as the potency of a drug.

  • High variation can lead to serious consequences, including potential harm due to excessive dosage or underdosing.

Sampling Distributions and Variance

  • When repeatedly sampling from a normal population with mean 𝜇 and standard deviation 𝜎, we can analyze the sampling distribution of variance (𝜎²).

Observing Variance Distribution Using R

  • R Code:

x=seq(1:10000)
for(i in 1:10000) { 
  x[i]=sd(rnorm(25))^2
}
boxplot(x,horizontal=T)

  • This code generates a boxplot of variances from 10,000 samples of normal distributions, each of size 25.

Distribution Characteristics

  • The distribution of variances is typically:

    • Right-skewed

    • Minimum value of 0

    • Maximum value is effectively infinite

  • This asymmetry indicates that we cannot apply normal or t-distributions as both are symmetric.

Chi-Square Distribution

  • The sampling distribution of the sample variance (s²) from a normal population is predictable and follows a chi-square distribution.

  • The formula involved is 𝜒² = (n - 1)s² / 𝜎², which follows a chi-square distribution with degrees of freedom df = n - 1.

Properties of the Chi-Square Distribution

  1. Shape

    • Right-skewed

  2. Value Range

    • Values can only be 0 or positive; negative values are not permissible.

  3. Degrees of Freedom

    • Different for each n (degrees of freedom) and influence the shape of the distribution.

  4. Mean and Variance

    • Mean (𝜇) equals df.

    • Variance (𝜎²) equals 2*df.

Visualization of Chi-Square Distribution

  • Displayed using various degrees of freedom (df) on a chart ranging from 0 to 25.

R Commands for Chi-Square Distribution

  • Probability function: pchisq(x, df, lower.tail=T)

  • Quantile function: qchisq(prob, df, lower.tail=T)

  • Density function: dchisq(x, df)

Assumptions for Chi-Square Tests

  • The primary assumption for utilizing the chi-square distribution is the existence of a random sample from a normal distribution.

  • Variance estimates based on this assumption enable hypothesis testing and confidence intervals.

Confidence Interval Estimates for Variance



  • The (1 - 𝛼) confidence interval estimate for 𝜎² is formulated as:(n − 1)s² / 𝜒²U, (n − 1)s² / 𝜒²Lwhere 𝜒²U and 𝜒²L are the respective critical chi-square values based on the given degrees of freedom and alpha level.

Chi-Square Test Procedure

  1. Hypothesis Formulation

    • Case 1: H0: 𝜎² ≤ 𝜎²₀

    • Case 2: H0: 𝜎² ≥ 𝜎²₀

    • Case 3: H0: 𝜎² = 𝜎²₀

  2. Significance Level (𝛼)

  3. Test Statistic Calculation

    • 𝜒² = (n - 1)s² / 𝜎²₀

  4. Decision Rule

    • Reject the null hypothesis based on the calculated chi-square values compared to critical values or p-values using designated significance levels.

Example in Practice: Orange Juice Container Monitoring

  • A production process monitors the amount of juice in 64-ounce containers with normal distribution with a mean = 64.3 ounces and standard deviation = 0.15 ounces, sampled with n = 24.

  • Data Analysis

  • Estimation of underfilled and overfilled containers focusing on variability (standard deviation).

Implementation Using R

  1. R Code for reading data:

x=read.table("ex7-5.txt",header=T,sep=",")
attempt(x)
ls(x)[1]  "ounces"
mean(ounces)
var(ounces)
boxplot(ounces, horizontal=TRUE)

Confidence Intervals and Hypothesis Tests for Variance

  • Construct confidence intervals and perform hypothesis testing for the variance, demonstrating practical implications.

  • Evaluate the statistical evidence regarding the population variance in practical scenarios.

End of Notes for Section 7.2

7.3 Estimation and Tests for Comparing Two Population Variances

Introduction to Comparison of Variances

  • Objective: To compare variances from two independent populations with respectively drawn samples n₁ and n₂ from normally distributed populations.

F-Distribution Overview

  • The ratio of variances s²₁ / s²₂ is determined using the F-distribution. The sampling distribution for this ratio reveals characteristics specific to population variances.

Key Properties of the F-Distribution

  • Can only assume positive values.

  • Right-skewed nature indicative of ratios.

  • Differentiation based on degrees of freedom specific for both variances (df₁ and df₂).

R Implementation for F-Distribution

  • Utilize R for commands:

  • Probability: pf(x, df1, df2, lower.tail=T)

  • Quantile: qf(prob, df1, df2, lower.tail=T)

  • Density: df(x, df1, df2)

Confidence Interval for Variances

  • Confidence interval estimates for the ratio of variances can be illustrated:

    • Utilizing critical values from F-distribution specific to calculated degrees of freedom.

Hypothesis Testing for Variance Ratios

  1. Hypotheses Setup

    • H0: 𝜎²₁ / 𝜎²₂ ≤ 1 vs Ha: 𝜎²₁ / 𝜎²₂ > 1

  2. Significance Level Determination (𝛼)

  3. Test Statistic Calculation

  4. Decision Rule Implementation Based on Test Statistic

Example Case: Socioeconomic Distributions of School Zones

  • Case study featuring sampling of family income across two attendance zones, verifying mean and variance differences.

  • Executing hypothesis tests using appropriate statistical programming in R to derive conclusions based on statistical significance.

Confidence Intervals and Statistical Conclusions

  • Establish confidence intervals for comparison of family incomes between zones, interpret conditions for validity of results, and validate procedures utilized.

Verification of Conditions for Testing

  • Confirm methodological rigor in testing and constructing confidence intervals, ensuring that requisite statistical assumptions are satisfied.

Conclusion

  • The analysis bears significance when addressing variances in production processes or empirical inquiry, substantiating the foundational principles of variance in statistics.