T-test Definition

• While the population’s standard deviation is unclear, the t-test is a statistical test employed to evaluate hypotheses about the mean of a small sample population.
• The t-test is utilized to determine whether there is a significant difference between the means of the two groups.
• To determine if a method impacts both samples or if the groups are different, the t-test is used.
• The t-test basically allows you to compare the means of two data sets and determine whether or not they were produced from the same population.
• The results generated by the t-test formulas are then compared to standard values once the null and alternative hypotheses have been established.
• According to the comparison, the null hypothesis is either accepted or rejected.
• With the exception of the fact that it is frequently used when the sample size is small, the T-test is similar to the z-test and the f-test (n30).

T-test Formula

T-tests could be carried out either automatically by software or manually by utilizing a formula.

The following formula may be used to manually calculate the t-value:

Where n is the number of observations; x denotes the sample mean; y denotes the assumed mean; and z denotes the standard deviation.

T-test for the Difference in Mean

where x̅1  and x̅2 are the mean of two samples and σ1 and σ2 is the standard deviation of two samples, and n1 and n2 are the numbers of observation of two samples.

T-test With one Sample (one-tailed t-test)

• The one-sample t-test is a statistical analysis in which the alternative hypothesis is supported if the population parameter is either more than or less than a predetermined value but not both.
• The alternative hypothesis must be taken instead of the null hypothesis if the sample’s t-score comes inside the critical range of a one-sided test.
• A one-tailed test is used to determine if the population is less than or more than a certain number.
• A one-tailed test is allowed if the estimated value can differ from the sample value in either the left or right direction but not both.
• The alternative hypothesis for this test states that the assumed value is either more than or less than the genuine mean but not both, whereas the null hypothesis states that there is no variation between the real mean and the assumed value.
• For instance, if H0: 0 = and Ha: 0, then the test is one-sided or, more accurately, left-tailed.
• There is just one rejection region on the left tail of the distribution under these parameters.
• If we assume = 100 and our sample mean deviates sufficiently from 100 in the negative direction, we reject H0 or the null hypothesis. If not, H0 is accepted at a particular significance level.
• Similarly, if H0 = 0 and Ha > 0, the rejection zone is visible on the right tail of the curve in this one-tailed test as well (right tail).
• When H0 equals 100, and the sample mean differs from 100 in an upward direction to a significant degree, H0 is denied; otherwise, it is permitted.

Two Sample t-test (two-tailed t-test)

• The two-sample t-test is a technique for determining if a sample’s population parameter exceeds or falls short of a predetermined range of values.
• Whenever the sample mean is significantly more or less than the predicted value of the population mean, a two-tailed test eliminates the null hypothesis.
• While the alternative hypothesis is a value that is not equal to the defined value of the null hypothesis and the null hypothesis is some assumed value, this type of test is appropriate.
• The two-tailed test is suitable when H0: = 0 and Ha: 0 are true, which might indicate > 0 or 0.
• In a two-tailed test, there are two rejection zones, one in each direction, left and right, toward the curve’s tails.
• Assuming = 100, the null hypothesis can be rejected if the sample mean deviates sufficiently from 100 in any way. However, if the sample mean is not significantly different from the null hypothesis is accepted.

Independent t-test

• An Independent t-test is a test used to compare the population means of two independent groups in order to establish the statistical evidence that the population means are substantially different.
• In addition, participants in each sample are supposed to originate from distinct populations, i.e., subjects in “Sample A” are assumed to come from “Population A,” while subjects in “Sample B” are assumed to come from “Population B.”
• It is assumed that the only difference between the populations is the level of the independent variable.
• As a result, any difference found between sample means and population means must also be present, as well as any variation between population means should be explained by the independent variable’s levels.
• This information may be used to build a curve that shows how an independent variable affects a dependent variable and vice versa.

T-test Example

Can a sample of 10 copper wires with a mean breaking strength of 527 kg be considered representative of a population with a mean breaking strength of 578 kg and a standard deviation of 12.72 kg? Test at a significance level of 5 percent.

Using the null hypothesis that the population’s mean breaking strength is 578 kg, we may write:

H0: µ = 578 kgs

Ha: µ ≠ 578 kgs

x̅ = 527 kgs, σ = 12.72, n = 10.

On the premise that the population is normally distributed, the formula for the test statistic t is:

t = (527+578) / (12.722/√10)

t = 21.597

As Ha is two-tailed in the above question, a two-tailed test has to be employed to determine the rejection areas at a significance level of 5%, which results in the following using the normal curve area table.

R: | t | > 1.96

The observed value of t is -1.488, which falls within the zone of acceptability given that R: | t | > 1.96; hence, H0 is accepted.

T-Test Applications

• The T-test is performed when comparing the means of two samples, dependent or independent.
• It may also be used to determine whether the sample mean deviates from the anticipated mean.
• To get the confidence interval for a sample mean, use the T-test.