Test Statistic Calculator – Loan calculator

Z-Test Statistic Calculator

Calculate the Z-statistic for a single population mean when the population standard deviation is known.

Sample Mean (x̄):

Hypothesized Population Mean (μ₀):

Population Standard Deviation (σ):

Sample Size (n):

Calculation Result:

"; resultDiv.innerHTML += "Z-Statistic: " + zStatistic.toFixed(4) + ""; resultDiv.innerHTML += "This Z-statistic indicates how many standard errors your sample mean (" + sampleMean + ") is away from the hypothesized population mean (" + hypothesizedMean + ")."; resultDiv.innerHTML += "You can compare this value to critical Z-values or use it to find a p-value to make a decision about your hypothesis."; }

Understanding the Test Statistic Calculator

In the realm of statistics, a test statistic is a numerical summary of a sample data set that is used to perform a hypothesis test. It quantifies how much the sample data deviates from what we would expect under the null hypothesis. By comparing this calculated test statistic to a critical value or using it to determine a p-value, we can make informed decisions about whether to reject or fail to reject the null hypothesis.

This calculator specifically computes the Z-statistic for a single population mean when the population standard deviation is known. This is a fundamental concept in inferential statistics, often used when you want to test if a sample mean is significantly different from a hypothesized population mean.

The Z-Test Statistic Formula

The formula for the Z-statistic in this context is:

Z = (x̄ - μ₀) / (σ / √n)

Where:

x̄ (x-bar) is the Sample Mean: The average value calculated from your collected sample data.
μ₀ (mu-naught) is the Hypothesized Population Mean: The value you are testing against, often derived from a null hypothesis (e.g., "the true population mean is 100").
σ (sigma) is the Population Standard Deviation: A measure of the spread or dispersion of the entire population. This value is assumed to be known for a Z-test.
n is the Sample Size: The number of observations or data points in your sample.

How to Interpret the Z-Statistic

Once you calculate the Z-statistic, its value tells you how many standard errors the sample mean (x̄) is away from the hypothesized population mean (μ₀).

A Z-statistic close to zero suggests that your sample mean is very close to the hypothesized population mean, providing little evidence against the null hypothesis.
A large positive or negative Z-statistic indicates that your sample mean is far from the hypothesized population mean, suggesting stronger evidence against the null hypothesis.

To make a formal decision, you would typically compare your calculated Z-statistic to critical values from a standard normal (Z) distribution table, or use it to find a p-value. If the absolute value of your Z-statistic exceeds the critical value (for a chosen significance level), or if your p-value is less than your significance level, you would reject the null hypothesis.

Example Scenario:

Imagine a company claims that the average weight of their product is 500 grams. You take a sample of 30 products and find their average weight to be 495 grams. From historical data, you know the population standard deviation of product weights is 10 grams. You want to test if the true average weight is indeed 500 grams.

Sample Mean (x̄): 495
Hypothesized Population Mean (μ₀): 500
Population Standard Deviation (σ): 10
Sample Size (n): 30

Using the calculator above, you would input these values to find the Z-statistic, which would then help you determine if the observed difference is statistically significant.

This calculator provides the first step in hypothesis testing by giving you the crucial test statistic.