Confidence Interval
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1
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%.
3
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.
6
Sample Variance (S²)
The variance calculated from a sample.
7
Sampling Distribution
The probability distribution of a statistic, such as the sample mean or sample variance, across repeated samples.
8
Degrees of Freedom (v)
The number of independent values available to estimate a statistical parameter.
9
Larger sample sizes
generally provide more reliable estimates of population parameters.
10
Smaller sample variance
generally indicates more consistent measurements.
11
Confidence intervals
account for sample size, variability, and the desired confidence level.
12
Chi-square (χ²)
Confidence interval for population variance
13
t (Student's t)
Confidence interval for population mean
14
F distribution
Confidence interval for the ratio of two population variances
15
Degrees of Freedom (v)
The number of independent pieces of information available in a sample.
16
v = n − 1
Where:
• v — Degrees of freedom.
• n — Number of observations.
Where:
• v — Degrees of freedom.
• n — Number of observations.
Degrees of Freedom
17
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.
18
Chi-Square Distribution
It compares the relationship between population variance (σ²) and sample variance (S²).
19
Chi-Square Distribution
It is an asymmetric distribution, meaning its left and right sides are not mirror images.
20
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
21
t Distribution
A probability distribution used to estimate the population mean using sample data.
22
t Distribution
It is similar to the normal distribution but accounts for additional uncertainty when the population standard deviation is unknown.
23
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.
25
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:
26
F Distribution
A probability distribution used to compare the variances of two sample sets.
27
F Distribution
It determines whether two sample variances may represent the same population variance.
28
F Distribution
It is commonly used in statistical testing and surveying network adjustments.
29
F Distribution
It is an asymmetric distribution.
30
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
31
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:
33
Confidence Interval for Population Variance: Chi-Square Test
Purpose: Determine the range within which the true population variance (σ²) is expected to lie.
34
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
35
Confidence Interval for Population Mean: t Statistics
Purpose: Determine the range within which the true population mean (μ) is expected to lie.
36
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
37
t distribution
accounts for greater uncertainty when estimating the population mean from a small sample.
38
Confidence Interval for the Ratio of Two Population Variances: F Test
Purpose: Determine whether two population variances are statistically equivalent.
39
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
40
Minimally Constrained Adjustment
Uses only the minimum control needed to fix the network.
41
Fully Constrained Adjustment
Uses all available control information.
42
Reference Variance
A measure of the variance of the residuals from an adjustment.
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