Confidence Interval
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Confidence Interval (CI)
A range of values within which the true population parameter is expected to lie at a specified confidence level.
2
Confidence Level
The percentage of confidence associated with an interval, such as 90%, 95%, or 99%.
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Population Mean (μ)
The average of all values in a population.
4
Sample Mean (ȳ)
The average of values obtained from a sample.
5
Population Variance (σ²)
The variance of the entire population.
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Sample Variance (S²)
The variance calculated from a sample.
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Sampling Distribution
The probability distribution of a statistic, such as the sample mean or sample variance, across repeated samples.
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Degrees of Freedom (v)
The number of independent values available to estimate a statistical parameter.
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Larger sample sizes
generally provide more reliable estimates of population parameters.
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Smaller sample variance
generally indicates more consistent measurements.
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Confidence intervals
account for sample size, variability, and the desired confidence level.
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Chi-square (χ²)
Confidence interval for population variance
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t (Student's t)
Confidence interval for population mean
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F distribution
Confidence interval for the ratio of two population variances
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Degrees of Freedom (v)
The number of independent pieces of information available in a sample.
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v = n − 1
Where:
• v — Degrees of freedom.
• n — Number of observations.
Where:
• v — Degrees of freedom.
• n — Number of observations.
Degrees of Freedom
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Chi-Square Distribution
A probability distribution used to determine the confidence interval of a population variance based on the sample variance and degrees of freedom.
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Chi-Square Distribution
It compares the relationship between population variance (σ²) and sample variance (S²).
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Chi-Square Distribution
It is an asymmetric distribution, meaning its left and right sides are not mirror images.
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χ² = (v × S²) / σ²
Where:
• χ² — Chi-square statistic.
• v — Degrees of freedom.
• S² — Sample variance.
• σ² — Population variance.
Where:
• χ² — Chi-square statistic.
• v — Degrees of freedom.
• S² — Sample variance.
• σ² — Population variance.
Chi-Square Distribution
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t Distribution
A probability distribution used to estimate the population mean using sample data.
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t Distribution
It is similar to the normal distribution but accounts for additional uncertainty when the population standard deviation is unknown.
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t Distribution
It is especially useful for small sample sizes, commonly fewer than 30 observations.
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t Distribution
It is symmetrical around zero.
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t = Z / √(χ² / v)
Where:
• t — Student's t random variable.
• Z — Standard normal random variable.
• χ² — Chi-square random variable.
• v — Degrees of freedom.
Where:
• t — Student's t random variable.
• Z — Standard normal random variable.
• χ² — Chi-square random variable.
• v — Degrees of freedom.
t Distribution Relationship with Other Distributions:
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F Distribution
A probability distribution used to compare the variances of two sample sets.
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F Distribution
It determines whether two sample variances may represent the same population variance.
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F Distribution
It is commonly used in statistical testing and surveying network adjustments.
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F Distribution
It is an asymmetric distribution.
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F = (χ₁² / v₁) / (χ₂² / v₂)
Where:
• F — F statistic.
• χ₁² — First chi-square random variable.
• χ₂² — Second chi-square random variable.
• v₁ — Numerator degrees of freedom.
• v₂ — Denominator degrees of freedom.
Where:
• F — F statistic.
• χ₁² — First chi-square random variable.
• χ₂² — Second chi-square random variable.
• v₁ — Numerator degrees of freedom.
• v₂ — Denominator degrees of freedom.
F Distribution
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S₁² / S₂² should be approximately 1.
If two sample variances represent the same population variance:
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F(α, v₁, v₂) = 1 / F(1 − α, v₂, v₁)
Lower-Tail Relationship:
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Confidence Interval for Population Variance: Chi-Square Test
Purpose: Determine the range within which the true population variance (σ²) is expected to lie.
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Lower Limit = (v × S²) / χ²(α/2)
Upper Limit = (v × S²) / χ²(1 − α/2)
Where:
• v — Degrees of freedom.
• S² — Sample variance.
• α — Significance level.
• χ² — Chi-square critical value.
Upper Limit = (v × S²) / χ²(1 − α/2)
Where:
• v — Degrees of freedom.
• S² — Sample variance.
• α — Significance level.
• χ² — Chi-square critical value.
Confidence Interval for Population Variance: Chi-Square Test
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Confidence Interval for Population Mean: t Statistics
Purpose: Determine the range within which the true population mean (μ) is expected to lie.
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Lower Limit = ȳ − t(α/2) × (S / √n)
Upper Limit = ȳ + t(α/2) × (S / √n)
Or:
CI = ȳ ± t(α/2) × (S / √n)
Where:
• ȳ — Sample mean.
• t — Critical t-value.
• S — Sample standard deviation.
• n — Number of observations.
• α — Significance level.
Upper Limit = ȳ + t(α/2) × (S / √n)
Or:
CI = ȳ ± t(α/2) × (S / √n)
Where:
• ȳ — Sample mean.
• t — Critical t-value.
• S — Sample standard deviation.
• n — Number of observations.
• α — Significance level.
Confidence Interval for Population Mean: t Statistics
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t distribution
accounts for greater uncertainty when estimating the population mean from a small sample.
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Confidence Interval for the Ratio of Two Population Variances: F Test
Purpose: Determine whether two population variances are statistically equivalent.
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Let R = S₁² / S₂².
Lower Limit = R / F(α/2, v₁, v₂)
Upper Limit = R × F(α/2, v₂, v₁)
Where:
• R — Ratio of sample variances.
• S₁² — First sample variance.
• S₂² — Second sample variance.
• v₁ — First degrees of freedom.
• v₂ — Second degrees of freedom.
• F — F-distribution critical value.
Lower Limit = R / F(α/2, v₁, v₂)
Upper Limit = R × F(α/2, v₂, v₁)
Where:
• R — Ratio of sample variances.
• S₁² — First sample variance.
• S₂² — Second sample variance.
• v₁ — First degrees of freedom.
• v₂ — Second degrees of freedom.
• F — F-distribution critical value.
Confidence Interval for the Ratio of Two Population Variances: F Test
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Minimally Constrained Adjustment
Uses only the minimum control needed to fix the network.
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Fully Constrained Adjustment
Uses all available control information.
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Reference Variance
A measure of the variance of the residuals from an adjustment.
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