2 Sample Z Test Calculator

Enter the required data to calculate with our 2 Sample Z Test Calculator that is both in basic mode and advanced calculator. Furthermore, read the formula and solved examples to get better understanding.

Formula:

$Z = \frac{X_1 – X_2}{\sqrt{\left(\frac{s_1^2}{n_1}\right) + \left(\frac{s_2^2}{n_2}\right)}}$

Variables

Variable	Meaning
$Z$	Z-score (difference in terms of standard deviations)
$X_1$	Mean of the first sample
$X_2$	Mean of the second sample
$s_1^2$	Variance of the first sample
$s_2^2$	Variance of the second sample
$n_1$	Size of the first sample
$n_2$	Size of the second sample

Solved Examples:

Example 1:

Given:

Mean of the first sample ( $X_1$ ) = 50
Mean of the second sample ( $X_2$ ) = 45
Variance of the first sample ( $s_1^2$ ) = 16
Variance of the second sample ( $s_2^2$ ) = 25
Size of the first sample ( $n_1$ ) = 30
Size of the second sample ( $n_2$ ) = 30

Calculation	Instructions
Step 1: Substitute the values into the formula:	Start with the formula.
$Z = \frac{50 – 45}{\sqrt{\left(\frac{16}{30}\right) + \left(\frac{25}{30}\right)}}$	Replace the variables with the given values.
Step 2: Calculate the components of the denominator:	Compute the variance ratios.
$\sqrt{\left(\frac{16}{30}\right) + \left(\frac{25}{30}\right)} \approx \sqrt{0.533 + 0.833}$	Simplify the fractions inside the square root.
Step 3: Add the values inside the square root:	Sum the simplified fractions.
$\sqrt{1.366} \approx 1.169$	Take the square root of the sum.
Step 4: Calculate the Z-score:	Divide the difference in means by the result from Step 3.
$Z = \frac{5}{1.169} \approx 4.28$	Final Z-score calculation.

Answer: The Z-score is approximately 4.28.

Example 2:

Given:

Mean of the first sample ( $X_1$ ) = 60
Mean of the second sample ( $X_2$ ) = 55
Variance of the first sample ( $s_1^2$ ) = 20
Variance of the second sample ( $s_2^2$ ) = 30
Size of the first sample ( $n_1$ ) = 40
Size of the second sample ( $n_2$ ) = 35

Calculation	Instructions
Step 1: Substitute the values into the formula:	Start with the formula.
$Z = \frac{60 – 55}{\sqrt{\left(\frac{20}{40}\right) + \left(\frac{30}{35}\right)}}$	Replace the variables with the given values.
Step 2: Calculate the components of the denominator:	Compute the variance ratios.
$\sqrt{\left(\frac{20}{40}\right) + \left(\frac{30}{35}\right)} \approx \sqrt{0.5 + 0.857}$	Simplify the fractions inside the square root.
Step 3: Add the values inside the square root:	Sum the simplified fractions.
$\sqrt{1.357} \approx 1.165$	Take the square root of the sum.
Step 4: Calculate the Z-score:	Divide the difference in means by the result from Step 3.
$Z = \frac{5}{1.165} \approx 4.29$	Final Z-score calculation.

Answer: The Z-score is approximately 4.29.

What is 2 Sample Z Test Calculator ?

The 2 Sample Z Test Calculator is a statistical tool used to compare the means of two independent samples to determine if there is a significant difference between them. This test is particularly useful when you want to know if the difference in sample means is large enough to be considered statistically significant.

The Formula calculates the Z-score, which tells you how many standard deviations the difference between the two means is from the expected difference (usually zero). This calculator is commonly used in fields such as medicine, economics, and social sciences to make informed decisions based on sample data.

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