# 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.

I should add that logistic regression, and dummy-coding the outcome, seems rather coarse for your purpose. You want to know how the timeline duration relates to the actual time it took for the lead to open an account. So it would probably be more informative not to dummy-code, and use a model that allows you to predict a lead's actual time based on their timeline duration. For example, you could try linear regression with the logarithm of the outcome variable.

• Thanks for the answer and sorry about not being more descriptive. I was running a linear regression model where I set the dependent variable as "Paying Accounts" and independent variable is "Within Timeline". I could be wrong (please let me know), but I was trying to see if the answer to "Timeline" is an accurate reflection of when we should expect the paying account date. Would you say that the inputs for my regression wouldn't deliver my answer as what you have proposed in your answer? – cphill Jul 27 '17 at 20:15
• @cphill I don't understand what you mean by "Paying Accounts". What does that variable represent? – Kodiologist Jul 27 '17 at 20:22
• Thank you for bearing with my lack of clarity :). I updated my question to show you the structure of my raw data and inputs for the regression. This data is related to a sales funnel, so the objective is to drive as many "Lead Date" records to also have a "Paying Account Date" within the sample. If they do, then a column called "Paying Accounts" that has a value of "1" to represent a "Paying Account Date" for that record. I am comparing the "Within Timeframe" to the "Paying Accounts" column – cphill Jul 27 '17 at 20:40
• @cphill But using Paying Accounts as the DV and Within Timeframe as the IV makes no sense because when Paying Accounts is 0, Within Timeframe is undefined. That is, if somebody didn't open an account, you can't meaningfully talk about when they opened it. – Kodiologist Jul 27 '17 at 21:36
• that is a valid point and I think not working with too much categorical information might have led me to believe something like my initial attempt was possible. In order to correct my regression based on the theory I originally proposed, would you remove those that haven't become a paying account from the sample or maybe use the approach you suggested in your original answer? Thank you for being so helpful so far! – cphill Jul 28 '17 at 14:24