C207 Task 1 Regression Business Question Example

Does a Warmer Winter Day Fill the Tunnel? Framing a Linear Regression Question for Staffing a Composite Express Car Wash Chain

Student Name

School of Business, Western Governors University

C207: Data-Driven Decision Making, Task 1

Course Instructor

Month Day, Year

Does a Warmer Winter Day Fill the Tunnel? Framing a Linear Regression Question for Staffing a Composite Express Car Wash Chain

The Business Situation

ClearLine Express, a composite chain of 14 express car washes in the upper Midwest, sells single washes and unlimited monthly memberships. Each site runs a conveyor tunnel and needs attendants to guide cars on, prep vehicles and sell memberships. Labor is the chain's largest controllable cost, about 31% of revenue. In winter, volume swings widely: on some days cars line up into the street, while on others the tunnel sits idle and attendants are sent home after the minimum shift guaranteed by company policy. Site managers set winter schedules a week in advance based on experience, and the regional director believes they overstaff cold days and understaff mild ones. The director has asked whether the weather forecast could be used to schedule staff more accurately.

The Business Question

The question is: during the winter months, how much does the number of cars washed per day at a ClearLine site change for each one-degree increase in the day's high temperature? The question suits linear regression because it asks about the size and direction of a relationship between two numeric variables and because the answer, a slope, translates directly into a staffing rule. If each degree adds a predictable number of cars, managers can convert a forecast into a staffing level. Managers already suspect that people wash their cars on milder days after snow and salt, but the chain does not know whether the effect is large enough to act on.

Variables

The dependent variable is daily wash volume: the total number of vehicles that passed through a site's tunnel in a day, including both single washes and member washes, recorded automatically by the tunnel controller. It is a count, measured in cars per day.

The independent variable is the day's observed high temperature, in degrees Fahrenheit, for the weather station nearest each site. Observed rather than forecast temperature is used for the first analysis, because the goal is to measure the underlying relationship; a second step would test how well forecasts predict volume.

Other factors also affect volume, including precipitation, day of the week and holidays. They are not included in this simple regression but will be recorded, so that the analysis can be extended to a multiple regression if the simple model leaves a large amount of unexplained variation.

Data and Sample

Wash volume data will come from the chain's point-of-sale and tunnel control systems, which store a record for every car. Temperature data will come from the National Oceanic and Atmospheric Administration's publicly available daily climate records for the station nearest each site (National Oceanic and Atmospheric Administration [NOAA], n.d.). The sample will include every day from December 1 through February 28 for the past three winters at all 14 sites, excluding days a site was closed for equipment failure. That yields roughly 3,700 site-days, more than enough for a stable estimate. Because sites differ in size, the first model will be run for the chain's three highest-volume sites separately, and the chain-wide model will use volume as a percentage of each site's winter average.

Hypotheses and Analysis Plan

Null hypothesis: in winter, there is no linear relationship between daily high temperature and daily wash volume; the slope of the regression line equals zero.

Alternative hypothesis: in winter, there is a linear relationship between daily high temperature and daily wash volume; the slope does not equal zero.

The analysis will fit a simple linear regression in Excel with wash volume as the dependent variable and high temperature as the independent variable, using a significance level of 0.05. Before trusting the result, the analyst will check the model's assumptions using a scatterplot for linearity, a residual plot for constant variance and a histogram of residuals for rough normality (Sharpe et al., 2019). If the scatterplot shows that volume rises with temperature only up to a point, for example because very mild days bring rain, the analyst will note the curve rather than force a straight line.

How the Results Would Be Interpreted and Used

Three numbers in the output will matter. The slope shows how many more or fewer cars a site washes for each additional degree; for example, a slope of 9 would mean about 9 more cars per degree. The p-value for the slope shows whether a relationship this strong would be unlikely if the true slope were zero. A p-value below 0.05 would lead the analyst to reject the null hypothesis, but statistical significance alone does not show that the effect is large enough to matter; the size of the slope and its confidence interval matter more for the decision (Wasserstein & Lazar, 2016). The R-squared value shows what share of the day-to-day variation in volume temperature explains; a value of 0.40, for example, would mean temperature explains a meaningful part of the variation but leaves most of it to other factors.

The decision rule follows from the result. If the slope is significant and large enough that a ten-degree swing changes volume by more than one attendant's workload, about 120 cars per shift, the chain will build a staffing table that sets each winter day's schedule from the forecast high. If the slope is significant but small, the chain will keep current staffing and look to other factors, such as snowfall. If there is no significant relationship, managers' belief will have been tested and rejected, which is useful in itself, because it stops the chain from building schedules around a pattern that is not there and points the next analysis toward the factors that do drive winter volume.

Limitations

A regression on past data shows association, not cause; warmer days may also be sunnier or follow snowstorms, and the model cannot separate those effects without more variables. Weather forecasts made a week ahead are less accurate than observed temperatures, so a staffing rule based on the model will perform less well than the model itself. And membership growth over three winters may shift volume independent of weather; including a time trend in a later model would address that.

References

National Oceanic and Atmospheric Administration. (n.d.). Climate data online. National Centers for Environmental Information. https://www.ncei.noaa.gov/cdo-web/

Sharpe, N. R., De Veaux, R. D., & Velleman, P. F. (2019). Business statistics (4th ed.). Pearson.

Wasserstein, R. L., & Lazar, N. A. (2016). The ASA statement on p-values: Context, process, and purpose. The American Statistician, 70(2), 129-133. https://doi.org/10.1080/00031305.2016.1154108