# Proper Phrasing of Hypothesis Test

I am running a linear regression on a set of data related to a sales funnel where I'm trying to determine the relationship between a date when a lead came in, the expected timeline for when they would open an account (user selected) and the actual date that they opened an account.

To better visualize:

A lead came in on a specific date, the user provided information that they would likely become a paying account within 15 days of when they provided their information (Lead date), in actuality they became a paying account before, on, or after that date.

• Timeline: 15 days
• Paying Account Date: 1/17/17
• Days Elapsed: 17 days
• Within (15 Day) Timeline?: No

For my sample of 7000 records, I created a dummy variable to convert the "Within Timeline" to a value (0 = No, 1 = Yes) based on if the paying account was generated before or after the 15 day timeframe.

From this I'm trying to determine the accurate phrasing of my hypothesis.

Here are two that I thought of, but not sure which is actually right based on my regression criteria:

1. Null = A users "Timeline" answer accurately reflects when a user becomes a paying account.

Alternate = A users "Timeline" answer does not accurately reflect when a user becomes a paying account.

2. Null = There is no relationship between a users "Timeline" answer and when a user becomes a paying account.

Alternate = There is a relationship between a users "Timeline" answer and when a user becomes a paying account.

Can anyone help me understand what an accurate hypothesis would be based on the criteria provided?

UPDATE:

Structure of my raw data:

• Column A = Lead created date (Date)
• Column B = Timeline (timeline duration, e.g. 15 days)
• Column C = Paying account date (Date)
• Column D = Paying Accounts (1 or 0 based on if record has value for Column C) (Dependent variable)
• Column E = Within Timeline (1 or 0 Based on Difference between Column A and C; compared to answer for Column B) (Independent variable)

You haven't fully specified the regression model in question, but I suppose you'd use a logistic-regression model of the form $\operatorname{logit} P(Y = 1) = β_0 + β_1X$, where $Y$ is Within Timeline and $X$ is the timeline duration in days. Then you would be concerned mostly about $β_1$, and the null hypothesis that $β_1 = 0$ could be interpreted as: the duration of a lead's timeline is unrelated to the probability of the lead opening an account within that timeline. The alternative hypothesis is that there is some nonzero (but still, possibly, arbitrarily small) association between these variables.