Lära z-test Means | Additional Tests

The z-test for means is a statistical method used to determine whether the means of two large, independent samples are significantly different from each other.

Key requirements:

Large sample sizes (typically n > 30 for each group);
Known population variances;
Independent samples.

The z-test is preferred over the t-test when the population variances are known and sample sizes are large. This situation is common in industrial or scientific settings where reliable historical data is available.

The z-test relies on the Central Limit Theorem, which states that the sampling distribution of the mean approaches a normal distribution as the sample size increases, regardless of the population's distribution. This means you can use the z-test when your data are approximately normally distributed, or when the sample size is large enough for the normal approximation to hold.

Test statistic:

The z-statistic for comparing two means is calculated as:

z = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}

Where:

$\bar{x}_1$ , $\bar{x}_2$ are the sample means;
$\sigma_{\raisebox{-1pt}{$1$}}^{\raisebox{1pt}{$2$}}$ , $\sigma_{\raisebox{-1pt}{$2$}}^{\raisebox{1pt}{$2$}}$ are the known population variances;
$n_1$ , $n_2$ are the sample sizes.

The resulting z-statistic measures how many standard errors the observed difference in means is away from the null hypothesis (usually that the means are equal). You then compare this value to the standard normal distribution to determine significance.


              12345678910111213141516171819202122
            
import numpy as np
from scipy.stats import norm

# Sample data: means, known variances, and sample sizes
mean1 = 105
mean2 = 100
var1 = 25  # Known variance of sample 1
var2 = 16  # Known variance of sample 2
n1 = 100   # Sample size 1
n2 = 120   # Sample size 2

# Calculate the standard error
se = np.sqrt(var1 / n1 + var2 / n2)

# Calculate the z-statistic
z = (mean1 - mean2) / se

# Calculate the two-tailed p-value
p_value = 2 * (1 - norm.cdf(abs(z)))

print(f"Z-statistic: {z:.2f}")
print(f"P-value: {p_value:.4f}")

Var allt tydligt?

Tack för dina kommentarer!

Avsnitt 2. Kapitel 1

Fråga AI

Fråga vad du vill eller prova någon av de föreslagna frågorna för att starta vårt samtal

Awesome!

Completion rate improved to 3.23

Svep för att visa menyn