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Z-Tests

Source: R/z_tests.R
z_tests.Rd

Statistical hypothesis tests for population means using the Normal (Z) distribution.

Z-tests are typically used when:

  1. The population variance is known.

  2. The sample size is large (N > 30), allowing the sample variance to approximate the population variance (via Central Limit Theorem).

One-Sample Tests:

  • one_sample_z_test: Performs a Z-test on a data vector u against an assumed_mean.

  • one_sample_z_test_params: Performs a Z-test given summary statistics (mean, variance, N).

Two-Sample Tests:

  • two_sample_z_test: Performs a Z-test comparing the means of two data vectors u and v.

Usage

one_sample_z_test_params(
  sample_mean,
  sample_variance,
  num_samples,
  assumed_mean
)

one_sample_z_test(u, assumed_mean)

two_sample_z_test(u, v)

Arguments

sample_mean

Numeric. The mean of the sample.

sample_variance

Numeric. The variance of the sample.

num_samples

Integer. The size of the sample.

assumed_mean

Numeric. The null hypothesis mean value to test against.

u

A numeric vector containing the first sample.

v

A numeric vector containing the second sample (for two-sample test).

Value

A numeric vector containing:

  1. Statistic: The computed Z-statistic.

  2. P-Value: The two-sided p-value associated with the Z-statistic.

Details

  • Statistic: The Z-statistic is calculated as $$Z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$$.

  • Assumptions: The underlying distribution is Normal, or the sample size is large enough for the CLT to apply.

See also

Boost Documentation

Examples

# --- One-Sample Z-Test ---
data1 <- c(5, 6, 7, 5, 6, 8)
# Test if population mean is 4
one_sample_z_test(data1, assumed_mean = 4)
#> [1] 4.539796923 0.006169843

# Using Summary Statistics
# Mean = 2, Variance = 1, N = 30, Null Mean = 0
one_sample_z_test_params(sample_mean = 2,
                         sample_variance = 1,
                         num_samples = 30,
                         assumed_mean = 0)
#> [1] 1.095445e+01 8.021287e-12

# --- Two-Sample Z-Test ---
data2 <- c(4, 5, 6, 4, 5, 7)
# Test if data1 and data2 have different means
two_sample_z_test(data1, data2)
#> [1] 1.4815944 0.1692572