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# Test Statistic Calculator

Choose the method, enter the values into the test statistic calculator, and click on the “Calculate” button to calculate the statistical value for hypothesis evaluation.

## Test Statistic Calculator

This test statistic calculator helps to find the static value for hypothesis testing. The calculated test value shows if there’s enough evidence to reject a null hypothesis. Also, this calculator performs calculations of either for one population mean, comparing two means, single population proportion, and two population proportions.

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

## What is Test Statistics?

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 Statistics Value?

• 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 = Mean of your sample data
• µ0 = Hypothesized population mean that you are comparing to your sample mean
• σ = Population standard deviation
• n = number of observations (sample size) in your data set

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 = 70 - 71 3√25

t = 1 0.6

t = 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.

$$\frac{√x̄ - √ȳ}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}$$

Where:

• ȳ = means of hypothesized 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 √5 2 / 30 + 6 2 / 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 t 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 = Population proportion

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 Proportion:

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:

• 1 and P̂2 = Sample proportions for two groups

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.

• 1 = 30 / 150 = 0.20
• 2 = 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