Z Stat Calculator with 2 Samples 90 Confidence Interval

This calculator helps you determine the z-statistic for comparing two sample means with a 90% confidence interval. The z-statistic measures how many standard errors a sample mean is from the population mean, helping you assess whether differences between samples are statistically significant.

What is a Z Statistic?

The z-statistic is a measure used in hypothesis testing to determine whether two population means are different when the variances are known and the sample size is large. It follows a standard normal distribution under the null hypothesis.

For two independent samples, the z-statistic is calculated as:

z = (x̄₁ - x̄₂) / √(σ₁²/n₁ + σ₂²/n₂)

Where:

x̄₁ and x̄₂ are the sample means
σ₁² and σ₂² are the population variances
n₁ and n₂ are the sample sizes

The z-statistic helps determine whether the difference between the two sample means is statistically significant. A larger absolute z-value indicates a greater likelihood that the difference is not due to random chance.

Two-Sample Z Test

The two-sample z-test compares the means of two independent samples to determine if they come from populations with the same mean. This test is appropriate when:

Both samples are independent
Population variances are known
Sample sizes are large (typically n > 30)

The test procedure involves:

Calculating the z-statistic using the formula above
Determining the critical z-value from standard normal distribution tables
Comparing the calculated z-value to the critical value
Making a decision about the null hypothesis

For a 90% confidence level, the critical z-values are approximately ±1.645.

90% Confidence Interval

A 90% confidence interval provides a range of values that is likely to contain the true population mean with 90% probability. For two samples, the confidence interval for the difference between means is calculated as:

(x̄₁ - x̄₂) ± z*(√(σ₁²/n₁ + σ₂²/n₂))

Where z* is the critical z-value for the desired confidence level (1.645 for 90%).

If the confidence interval does not include zero, it suggests a statistically significant difference between the two sample means at the 90% confidence level.

How to Use This Calculator

To use this calculator:

Enter the sample means (x̄₁ and x̄₂)
Enter the population standard deviations (σ₁ and σ₂)
Enter the sample sizes (n₁ and n₂)
Click "Calculate" to see the z-statistic and confidence interval

The calculator will display:

The calculated z-statistic
The 90% confidence interval for the difference between means
A visual representation of the confidence interval

Note: This calculator assumes known population variances. For unknown variances, use a t-test instead.

Worked Example

Suppose we have two samples:

Sample 1: Mean = 72, Standard Deviation = 8, Size = 50
Sample 2: Mean = 68, Standard Deviation = 10, Size = 60

Calculating the z-statistic:

z = (72 - 68) / √((8²/50) + (10²/60)) z = 4 / √(1.28 + 1.6667) z = 4 / √2.9467 z ≈ 4 / 1.7166 z ≈ 2.33

The 90% confidence interval for the difference between means is approximately:

(72 - 68) ± 1.645*(√(1.28 + 1.6667)) 4 ± 1.645*1.7166 4 ± 2.82 (1.18, 6.82)

This means we are 90% confident that the true difference in means lies between 1.18 and 6.82.

Frequently Asked Questions

What is the difference between a z-test and a t-test?: A z-test is used when population variances are known, while a t-test is used when variances are unknown. This calculator assumes known variances, so a z-test is appropriate.
When should I use a 90% confidence level instead of 95%?: A 90% confidence level is appropriate when you need more conservative results, or when you have limited sample sizes. It provides a wider confidence interval that is less likely to be affected by sampling error.
What does it mean if the confidence interval includes zero?: If the confidence interval includes zero, it suggests that the difference between the two sample means is not statistically significant at the 90% confidence level. This means you cannot conclude that there is a real difference between the populations.
Can I use this calculator for small sample sizes?: This calculator is designed for large sample sizes (n > 30). For small samples, you should use a t-test instead, as the z-distribution becomes less accurate with small sample sizes.
How do I interpret a negative z-statistic?: A negative z-statistic simply indicates that the first sample mean is lower than the second sample mean. The absolute value of the z-statistic is what matters for determining statistical significance.