# t, F, and ρ

There are a few things to remember,

Parameter | Shorthand | Description |
---|---|---|

Significance level | For a 95% level of confidence, we calculate our For a one-tailed test, For a two-tailed test, | |

Test statistic | The test statistic must be calculated. Once we have the test statistic, we can immediately test our hypothesis against the critical values. Alternatively, we can use it to calculate a And recall, of course, that | |

Critical value | With this, we can examine how our test statistic (t) performs with our significance level. The t-distribution table, a.k.a. a critical value distribution table, gives us the t-values we would need to see for various significance levels and degrees of freedom, if we are to accept or reject or hypothesis. Essentially, we reject the null hypothesis if the test statistic "exceeds" the critical value. However, that can change depending on what test we are performing. For a two-sided test, we reject the null hypothesis if the absolute value of the test statistic is greater than the critical value. For a one-sided upper test, we reject the null hypothesis if the test statistic is greater than the critical value. For a one-sided lower test, we reject the null hypothesis if the test statistic is less than the negative of the critical value.
| |

Probability value | The probability value calculates the likelihood that our observed t-statistic would exceed the critical value, thus rejecting the null hypothesis. It calculated using the t-statistic, and the degrees of freedom. Alternatively, a table is The If the | |

F | * |

## Hypothesis testing

Two-sided test

Using the test statistic,

If the absolute value of the test statistic is greater than the critical value, we reject the null hypothesis.

If the absolute value of the test statistic is less than the critical value, we accept the null hypothesis.

Using the probability value,

If the probability value is less than the significance level

, we reject the null hypothesis. If the probability value is greater than the significance level

, we accept the null hypothesis.

One-sided upper test

Using the test statistic,

If the test statistic is greater than the critical value, we reject the null hypothesis.

If the test statistic is less than the critical value, we accept the null hypothesis.

Using the probability value,

Same as two-sided test.

One-sided lower test

Using the test statistic,

If the statistic is less than the negative of the critical value, we reject the null hypothesis.

If the test statistic is greater than the negative of the critical value, we accept the null hypothesis.

Using the probability value,

Same as two-sided test.

## Critical values table for the t-distribution

Using spreadsheet software, the critical value for a particular significance level (

Degrees of freedom | One tail: 90% | One tail: 95.0% | One tail: 97.50% | One tail: 99.0% | One tail: 99.50% | One tail: 99.90% | One tail: 99.950% |
---|---|---|---|---|---|---|---|

0.05 | 0.025 | 0.01 | 0.005 | 0.001 | 0.0005 | ||

1 | 3.07768 | 6.31375 | 12.70620 | 31.82052 | 63.65674 | 318.30884 | 636.61925 |

2 | 1.88562 | 2.91999 | 4.30265 | 6.96456 | 9.92484 | 22.32712 | 31.59905 |

3 | 1.63774 | 2.35336 | 3.18245 | 4.54070 | 5.84091 | 10.21453 | 12.92398 |

4 | 1.53321 | 2.13185 | 2.77645 | 3.74695 | 4.60409 | 7.17318 | 8.61030 |

5 | 1.47588 | 2.01505 | 2.57058 | 3.36493 | 4.03214 | 5.89343 | 6.86883 |

6 | 1.43976 | 1.94318 | 2.44691 | 3.14267 | 3.70743 | 5.20763 | 5.95882 |

7 | 1.41492 | 1.89458 | 2.36462 | 2.99795 | 3.49948 | 4.78529 | 5.40788 |

8 | 1.39682 | 1.85955 | 2.30600 | 2.89646 | 3.35539 | 4.50079 | 5.04131 |

9 | 1.38303 | 1.83311 | 2.26216 | 2.82144 | 3.24984 | 4.29681 | 4.78091 |

10 | 1.37218 | 1.81246 | 2.22814 | 2.76377 | 3.16927 | 4.14370 | 4.58689 |

11 | 1.36343 | 1.79588 | 2.20099 | 2.71808 | 3.10581 | 4.02470 | 4.43698 |

12 | 1.35622 | 1.78229 | 2.17881 | 2.68100 | 3.05454 | 3.92963 | 4.31779 |

13 | 1.35017 | 1.77093 | 2.16037 | 2.65031 | 3.01228 | 3.85198 | 4.22083 |

14 | 1.34503 | 1.76131 | 2.14479 | 2.62449 | 2.97684 | 3.78739 | 4.14045 |

15 | 1.34061 | 1.75305 | 2.13145 | 2.60248 | 2.94671 | 3.73283 | 4.07277 |

16 | 1.33676 | 1.74588 | 2.11991 | 2.58349 | 2.92078 | 3.68615 | 4.01500 |

17 | 1.33338 | 1.73961 | 2.10982 | 2.56693 | 2.89823 | 3.64577 | 3.96513 |

18 | 1.33039 | 1.73406 | 2.10092 | 2.55238 | 2.87844 | 3.61048 | 3.92165 |

19 | 1.32773 | 1.72913 | 2.09302 | 2.53948 | 2.86093 | 3.57940 | 3.88341 |

20 | 1.32534 | 1.72472 | 2.08596 | 2.52798 | 2.84534 | 3.55181 | 3.84952 |

21 | 1.32319 | 1.72074 | 2.07961 | 2.51765 | 2.83136 | 3.52715 | 3.81928 |

22 | 1.32124 | 1.71714 | 2.07387 | 2.50832 | 2.81876 | 3.50499 | 3.79213 |

23 | 1.31946 | 1.71387 | 2.06866 | 2.49987 | 2.80734 | 3.48496 | 3.76763 |

24 | 1.31784 | 1.71088 | 2.06390 | 2.49216 | 2.79694 | 3.46678 | 3.74540 |

25 | 1.31635 | 1.70814 | 2.05954 | 2.48511 | 2.78744 | 3.45019 | 3.72514 |

26 | 1.31497 | 1.70562 | 2.05553 | 2.47863 | 2.77871 | 3.43500 | 3.70661 |

27 | 1.31370 | 1.70329 | 2.05183 | 2.47266 | 2.77068 | 3.42103 | 3.68959 |

28 | 1.31253 | 1.70113 | 2.04841 | 2.46714 | 2.76326 | 3.40816 | 3.67391 |

29 | 1.31143 | 1.69913 | 2.04523 | 2.46202 | 2.75639 | 3.39624 | 3.65941 |

30 | 1.31042 | 1.69726 | 2.04227 | 2.45726 | 2.75000 | 3.38518 | 3.64596 |

60 | 1.29582 | 1.67065 | 2.00030 | 2.39012 | 2.66028 | 3.23171 | 3.46020 |

120 | 1.28865 | 1.65765 | 1.97993 | 2.35782 | 2.61742 | 3.15954 | 3.37345 |

1000 | 1.28240 | 1.64638 | 1.96234 | 2.33008 | 2.58075 | 3.09840 | 3.30028 |

1.28155 | 1.64485 | 1.95996 | 2.32635 | 2.57583 | 3.09023 | 3.29053 |

table, or P table

This table was calculated using the spreadsheet function with inputs for the t-statistic (t), degrees of freedom (DF), and whether our test is two-sided (2), or one-sided (1): TDIST(t, DF, sidedness). The resulting

Row: t-statistic | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

1.30 | 0.209 | 0.162 | 0.142 | 0.132 | 0.125 | 0.121 | 0.117 | 0.115 | 0.113 | 0.111 |

1.32 | 0.206 | 0.159 | 0.139 | 0.129 | 0.122 | 0.117 | 0.114 | 0.112 | 0.110 | 0.108 |

1.34 | 0.204 | 0.156 | 0.136 | 0.126 | 0.119 | 0.114 | 0.111 | 0.109 | 0.107 | 0.105 |

1.36 | 0.202 | 0.153 | 0.134 | 0.123 | 0.116 | 0.111 | 0.108 | 0.105 | 0.103 | 0.102 |

1.38 | 0.200 | 0.151 | 0.131 | 0.120 | 0.113 | 0.108 | 0.105 | 0.102 | 0.100 | 0.099 |

1.40 | 0.197 | 0.148 | 0.128 | 0.117 | 0.110 | 0.106 | 0.102 | 0.100 | 0.098 | 0.096 |

1.42 | 0.195 | 0.146 | 0.125 | 0.114 | 0.107 | 0.103 | 0.099 | 0.097 | 0.095 | 0.093 |

1.44 | 0.193 | 0.143 | 0.123 | 0.112 | 0.105 | 0.100 | 0.097 | 0.094 | 0.092 | 0.090 |

1.46 | 0.191 | 0.141 | 0.120 | 0.109 | 0.102 | 0.097 | 0.094 | 0.091 | 0.089 | 0.087 |

1.48 | 0.189 | 0.139 | 0.118 | 0.106 | 0.099 | 0.095 | 0.091 | 0.089 | 0.087 | 0.085 |

1.50 | 0.187 | 0.136 | 0.115 | 0.104 | 0.097 | 0.092 | 0.089 | 0.086 | 0.084 | 0.082 |

1.52 | 0.185 | 0.134 | 0.113 | 0.102 | 0.094 | 0.090 | 0.086 | 0.083 | 0.081 | 0.080 |

1.54 | 0.183 | 0.132 | 0.111 | 0.099 | 0.092 | 0.087 | 0.084 | 0.081 | 0.079 | 0.077 |

1.56 | 0.181 | 0.130 | 0.108 | 0.097 | 0.090 | 0.085 | 0.081 | 0.079 | 0.077 | 0.075 |

1.58 | 0.180 | 0.127 | 0.106 | 0.095 | 0.087 | 0.083 | 0.079 | 0.076 | 0.074 | 0.073 |

1.60 | 0.178 | 0.125 | 0.104 | 0.092 | 0.085 | 0.080 | 0.077 | 0.074 | 0.072 | 0.070 |

1.62 | 0.176 | 0.123 | 0.102 | 0.090 | 0.083 | 0.078 | 0.075 | 0.072 | 0.070 | 0.068 |

1.64 | 0.174 | 0.121 | 0.100 | 0.088 | 0.081 | 0.076 | 0.073 | 0.070 | 0.068 | 0.066 |

1.66 | 0.173 | 0.119 | 0.098 | 0.086 | 0.079 | 0.074 | 0.070 | 0.068 | 0.066 | 0.064 |

## F-statistic, a.k.a F-multiplier

While the t-statistic is easily calculated based on the percentage of the interval and the degrees of freedom (derived from the sample or population size), the F-statistic has two sets of degrees of freedom (DF). There is the

The numerator will generally be set as

So for example, if we want are using 95% as our confidence, and we are working with 100 samples for two variables, then we wind up with

The spreadsheet command is,

=F.DIST.RT(x, degree_freedom1, degree_freedom2)

The spreadsheet command for the F critical value is,

=F.INV.RT(

## See it in action

Please refer to this Google Sheets spreadsheet,

https://docs.google.com/spreadsheets/d/1H3EtaltideRpUeVNMq7jxO2mea8NGcXHz4bYxhAJu58/edit?usp=sharing

## Endnotes

https://online.stat.psu.edu/stat200/book/export/html/213

Some reading about the p value,

https://www.math.arizona.edu/~piegorsch/571A/TR194.pdf

A useful reference table for F values is below,

http://www.socr.ucla.edu/Applets.dir/F_Table.html

How to calculate the p-value in spreadsheet software,

https://support.google.com/docs/answer/3295914?hl=en

https://spreadsheeto.com/p-value-excel/#p-value

https://ms-office.wonderhowto.com/how-to/find-p-value-with-excel-346366/

Some reading about the various tables

https://www.sheffield.ac.uk/polopoly_fs/1.43999!/file/tutorial-10-reading-tables.pdf