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# What is a Standardized Test Statistic?

AÂ statistical hypothesisÂ is an assumption about aÂ population parameter. For example, we may assume that the mean height of a male in the U.S. is 70 inches. The assumption about the height is theÂ statistical hypothesisÂ and the true mean height of a male in the U.S. is theÂ population parameter.

AÂ hypothesis test is a formal statistical test we use to reject or fail to reject some statistical hypothesis.

The basic process for performing a hypothesis test is as follows:

1. Collect sample data.

2. Calculate the standardized test statistic for the sample data.

3. Compare the standardized test statistic to some critical value. If itâ€™s more extreme than the critical value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis test.

The formula that we use to calculate theÂ standardized test statisticÂ varies depending on the type of hypothesis test we perform.

The following table shows the formula to use to calculate the standardized test statistic for each of the four major types of hypothesis tests:

### Hypothesis Test for One Mean

AÂ one sample t-testÂ is used to test whether or not the mean of a population is equal to some value.

The standardized test statistic for this type of test is calculated as follows:

t = (xÂ â€“ Î¼) / (s/âˆšn)

where:

• x:Â sample mean
• Î¼0:Â hypothesized population mean
• s:Â sample standard deviation
• n:Â sample size

Refer to this tutorial for an example of how to calculate this standardized test statistic.

### Hypothesis Test for a Difference in Means

AÂ two sample t-testÂ is used to test whether or not the means of two populations are equal.

The standardized test statistic for this type of test is calculated as follows:

t = (x1Â â€“Â x2)Â  /Â  sp(âˆš1/n1Â + 1/n2)

whereÂ x1Â andÂ x2Â are the sample means,Â n1Â and n2Â are the sample sizes, and whereÂ spÂ is calculated as:

spÂ =Â âˆšÂ (n1-1)s12Â +Â Â (n2-1)s22Â /Â  (n1+n2-2)

where s12Â and s22Â are the sample variances.

Refer to this tutorial for an example of how to calculate this standardized test statistic.

### Hypothesis Test for One Proportion

AÂ one proportion z-testÂ is used to compare an observed proportion to a theoretical one.

The standardized test statistic for this type of test is calculated as follows:

z = (p-p0) / âˆšp0(1-p0)/n

where:

• p:Â observedÂ sample proportion
• p0:Â hypothesized population proportion
• n:Â sample size

Refer to this tutorial for an example of how to calculate this standardized test statistic.

### Hypothesis Test for a Difference in Proportions

AÂ two proportion z-testÂ is used to test for a difference between two population proportions.

The standardized test statistic for this type of test is calculated as follows:

z =Â (p1-p2) / âˆšp(1-p)(1/n1+1/n2)

where p1Â and p2 are the sample proportions,Â n1 and n2Â are the sample sizes, and where p is the total pooled proportionÂ calculated as:

p = (p1n1 + p2n2)/(n1+n2)

Refer to this tutorial for an example of how to calculate this standardized test statistic.