Editing Logistic regression (section)

===Problem===
As a simple example, we can use a logistic regression with one explanatory variable and two categories to answer the following question:
<blockquote>
A group of 20 students spends between 0 and 6 hours studying for an exam. How does the number of hours spent studying affect the probability of the student passing the exam?
</blockquote>

The reason for using logistic regression for this problem is that the values of the dependent variable, pass and fail, while represented by "1" and "0", are not [[cardinal number]]s. If the problem was changed so that pass/fail was replaced with the grade 0–100 (cardinal numbers), then simple [[regression analysis]] could be used.

The table shows the number of hours each student spent studying, and whether they passed (1) or failed (0).
{| class="wikitable"
|-
! Hours (''x<sub>k</sub>'')
| 0.50|| 0.75|| 1.00|| 1.25|| 1.50|| 1.75|| 1.75|| 2.00|| 2.25|| 2.50|| 2.75|| 3.00|| 3.25|| 3.50|| 4.00|| 4.25|| 4.50|| 4.75|| 5.00 || 5.50
|-
! Pass (''y<sub>k</sub>'')
| 0|| 0|| 0|| 0|| 0|| 0|| 1|| 0|| 1|| 0|| 1|| 0|| 1|| 0|| 1|| 1|| 1|| 1|| 1|| 1
|}

We wish to fit a logistic function to the data consisting of the hours studied (''x<sub>k</sub>'') and the outcome of the test (''y<sub>k</sub>''&nbsp;=1 for pass, 0 for fail). The data points are indexed by the subscript ''k'' which runs from <math>k=1</math> to <math>k=K=20</math>. The ''x'' variable is called the "[[explanatory variable]]", and the ''y'' variable is called the "[[categorical variable]]" consisting of two categories: "pass" or "fail" corresponding to the categorical values 1 and 0 respectively.