Students often go straight to the hypothesis test rather than investigating the data with summary statistics and charts first. Encourage them to summarise their data first. As well as summarising their results, charts especially can show outliers and patterns.

For continuous normally distributed data, summarise using means and standard deviations. If the data is skewed or there are influential outliers, the median (middle value) and interquartile range (Upper quartile – lower quartile) are more appropriate.

T-tests are of different types:-

- Paired T-test – dependent and independent.
- Normal T-test

The paired t-test is used to determine paired differences. It is used in the cases where the sample is less than 50 and the sample on which the test was priory applied remains the same.

The One-sample t-test is used to compare a sample mean to a specific value.

t = (mean – comparison value)/ Standard Error

An “F Test” uses the F-distribution. It uses an F Statistic to compare two variances.

i.e. s_{1} and s_{2}, by dividing them. A result is always a number greater than zero (as variances are always positive). The equation for comparing two variances with the f-test is:

F = s^{2}_{1} / s^{2}_{2}

It is also essential to understand the difference between a t-test and f-test as they are using interchangeably by many people.

**T-test vs F-test**

The difference between the t-test and f-test is that t-test is used to test the hypothesis whether the given mean is significantly different from the sample mean or not. On the other hand, an F-test is used to compare the two standard deviations of two samples and check the variability.

## Comparison Table

Parameter of Comparison | T-test | F-test |
---|---|---|

Implication | The T-test is used to test the hypothesis whether the given mean is significantly different from the sample mean or not | F-test is used to compare the two standard deviations of two samples and check the variability. An F-test is a ratio of two Chi-squares. |

Types | T-tests are of different types:- 1. Paired T-test – dependent and independent. 2. Normal T-test | There is one type if F-test which is used to compare standard deviations of the two-sample data. |

Null Hypothesis | H0: the sample mean is equal to 0. | H0: the two samples have the same variance. |

Test statistic | T = (mean – comparison value)/ Standard Error ~t(n-1) | F = s21 / s22 ~ F(n1-1,n2-1) |

Degree of freedom | The degree of freedom is )n-1) where n is the number of sample values | The degree of freedom is (n1-1,n2-1) where n1 and n2 are the numbers of observations in samples 1 and 2. |

## What is T-test?

**T** distribution or t-test is used when the sample size,n, is less than 30 and the standard deviation, sigma, is unknown.

The distribution of continuous data can often be closely approximated by the normal distribution.

T distribution is generally used to calculate numerical data.it is derived from a normal distribution and is also just a type of normal distribution.

__One Sample t-test__

__One Sample t-test__

The one-sample t-test is concerned with making inferences regarding a population mean.

One sample t-test is used when we are given with only one sample and we need to run a hypothesis on that sample itself.

**Two Sample t-test**

**Two Sample t-test**

This is more common in a scenario than the one-sample t-test. Usually, we want to compare the means of 2 groups.

Two sample t-test is also used when we are given with only one sample and we need to run a hypothesis on that sample itself.

We can run two types of tests under this category.

- Paired test:- in this the same sample population is used for testing two different treatments. compare the means of two conditions in which the same (or closely matched) participants participated.
- Unrelated Samples:- In this, we compare the means of two groups of participants.

**Hypothesis testing with t **

- We can draw a sampling distribution of t-values (the Student t distribution) – this shows the likelihood of each t-value if the null hypothesis is true
- The distribution will be affected by sample size (or more precisely, by degrees of freedom)
- We evaluate the likelihood of obtaining our t-value given the t-distribution.

**Assumptions **

The one-sample t-test requires the following statistical assumptions:

- Random and independent sampling.
- Data are from normally distributed populations.

[Note: The one-sample t-test is generally considered robust against violation of this assumption once N > 30.]

## What is F-test?

An “F Test” uses the F-distribution. It uses an F Statistic to compare two variances.

F-test for detecting the identity of variances of two normally distributed random variables:-

Our hypothesis for the identity of the variances of two independent random variables of a normal distribution with unknown expectation and variance is checked by the so-called F-test.

H0: σ_{1}^{2} = σ_{2}^{2}

H1: σ_{1}^{2} > σ_{2}^{2}

The test is always carried out as a one-sided test.

Test statistics: F_{sz} = s_{1}^{2}/s_{2}^{2} where s_{1}^{2} > s_{2}^{2}

If H0 fulfills, then Fsz is of F-distribution with degrees of freedom n1-1, n2-1 .

Decision principle: for Fsz ≤ Fα 0-hypothesis is accepted, otherwise not.

**Main Differences Between T-test and F-test**

- The
**main difference between Reference and Recommendation**is, that t-test is used to test the hypothesis whether the given mean is significantly different from the sample mean or not. On the other hand, an F-test is used to compare the two standard deviations of two samples and check the variability. - The T-test can be conducted the two-sided test or a single-sided test but the f-test is the only single-sided test as variance can not be negative.
- T-tests are of different types:- Paired T-test – dependent and independent, Normal T-test. Whereas the f-test is only of one type.
- The T-test is applied when the sample population is less than 30 and the standard deviation is unknown, whereas the f-test can be applied upon the large sampled population.
- The T-test is used to check the hypothesis for the sample mean whereas the f-test is used to run the hypothesis on the variance of the samples.

## References

- https://asa.scitation.org/doi/abs/10.1121/1.417933
- https://projecteuclid.org/euclid.aoms/1177728261
- https://www.mitpressjournals.org/doi/abs/10.1162/089976699300016007

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