Test Statistic Calculator

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Test Statistic Calculator

This test statistic calculator helps determine the test value for hypothesis testing. The calculated test value indicates whether there’s enough evidence to reject the null hypothesis. It works for:

One population mean
Two population means
Single population proportion
Two population proportions

Our t statistic calculator is highly useful in various fields like research, experimentation, quality control, and data analysis.

What is a Test Statistic?

A test statistic is a numerical value obtained from the sample data set. It summarizes the differences between what you observe within your sample and what would be expected if a hypothesis were true.

The t-test statistic also shows how closely your data matches the predicted distribution among the sample tests you perform.

How to Calculate Test Statistic Value?

Understanding how to find test statistics is essential when analyzing data to determine whether to reject a null hypothesis in hypothesis testing. For this process, here are some steps given to find how far the observed data deviates from what is expected under the null hypothesis.

Collect the data from the populations
Use the data to find the standard deviation of the population
Calculate the mean (μ) of the population using this data
Determine the z-value or sample size
Use the suitable test statistic formula and get the results

Test Statistic for One Population Mean

Test statistics for a single population mean is calculated when a variable is numeric and involves one population or a group.

x̄ - μ₀ σ / √n

Where:

x̄ = Sample mean
μ₀ = Hypothesized population mean
σ = Population standard deviation
n = Sample size

Example:

Suppose we want to test if the average height of adult males in a city is 70 inches. We take a sample of 25 adult males and find the sample mean height to be 71 inches with a sample standard deviation of 3 inches. We use a significance level of 0.05.

t = (71 - 70) (3 / √25)

t = 1 0.6

≈ 1.67

Test Statistic Comparing Two Population Means

This test is applied when the numeric value is compared across the various populations or groups. To compute the resulting t statistic, two distinct random samples must be chosen, one from each population.

$$\frac{\sqrt{\bar{x}}-\sqrt{\bar{y}}}{\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}}$$

Where:

ȳ = Mean of the second sample or population

Example:

Suppose we want to test if there is a difference in average test scores between two schools. We take a sample of 30 students from school A with an average score of 85 and a standard deviation of 5, and a sample of 35 students from school B with an average score of 82 and a standard deviation of 6.

t = 85 - 82 √5² / 30 + 6² / 35

t = 3 √ 25/30 + 36/35

t = 3 √0.833 + 1.029

t = 3 √1.862

t = 2.20

Test Statistic for a Single Population Proportion

This test is used to determine if a single population's proportion differs from a specified standard. The standardized test statistic calculator works for a population proportion when dealing with data by having a limit of P₀ because proportions represent parts of a whole and cannot logically exceed the total or be negative.

$$\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$$

Where:

P̂ = Sample proportion
P₀ = Hypothesized population proportion
n = Sample size

Example:

Suppose we want to test if the proportion of left-handed people in a population is 10%. We take a sample of 100 people and find that 8 are left-handed. We use a significance level of 0.05.

= P̂ - P₀ √0.10 (1 - 0.10)/100

= 0.08 - 0.10 √0.10 (1 - 0.10)/100

= -0.02 √0.10 (0.9)/100

= -0.02 √0.009

= -0.02 0.03

= −0.67

Test Statistic for Two Population Proportions

This test identifies the difference in proportions between two independent groups to assess their significance.

$$\frac{\hat{p}_1-\hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1}+\frac{1}{n_2})}}$$

Where:

P̂₁, P̂₂ = Sample proportions of groups 1 and 2
P̂ = Combined proportion of successes across both samples

Example:

Suppose we want to test if the proportion of smokers is different between two cities. We take a sample of 150 people from City A and find that 30 are smokers, and a sample of 200 people from City B and find that 50 are smokers.

P̂1 = 30 / 150 = 0.20
P̂2 = 50 / 200 = 0.25
P̂ = 30 + 50 / 150 + 200 = 0.229

Calculation:

= 0.20 - 0.25 √0.229 (1 - 0.229) (1 / 150 + 1/200)

= -0.05 √0.229 (0.771) (1 / 150 + 1 / 200)

= -0.05 √0.176 (1/150 + 1/200)

= -0.05 √0.176 (0.0113)

= -0.05 √0.002

= -0.05 0.045

= −1.11