Performance Lawn Equipment $($PLE$),$ headquartered in St$.$ Louis, Missouri, is a privately owned designer and producer of traditional lawn mowers used by homeowners. In the past $10$ years, PLE has added another key product, a medium$-$size diesel power lawn tractor with front and rear power takeoffs, Class I three$-$point hitches, four$-$wheel drive, power steering, and full hydraulics. This equipment is built primarily for a niche market consisting of large estates, including golf and country clubs, resorts, private estates, city parks, large commercial complexes, resorts, private estates, city parks, lawn care service providers, etc. PLE provides most of the products to dealerships, which, in turn, sell directly to end users. PLE employs $1,660$ people worldwide. About half the workforce is based in St$.$ Louis; the remainder is split among their manufacturing plants.

In the United States, the focus of sales is on the eastern seaboard, California, the Southeast, and the south$-$central states, which have the greatest concentration of customers. Outside the United States, PLE$$s sales include a European market, a growing South American market, and developing markets in the Pacific Rim and China. The market is cyclical, but the different products and regions balance some of this, with just less than $30\%$ of total sales in the spring and summer $($in the United States$),$ about $25\%$ in the fall, and about $20\%$ in the winter. Annual sales are approximately $$180$ million.

PLE has several key suppliers: Mitsitsiu, Inc., the sole source of all diesel engines; LANTO Axles, Inc., which provides tractor axles; Schorst Fabrication, which provides subassemblies; Cuberillo, Inc., a supplier of transmissions; and Specialty Machining, Inc., a supplier of precision machine parts.

Elizabeth Burke has recently joined the PLE management team to oversee production operations. In reviewing data from the company$$s data warehouse $/$ data mart, she noticed that defects received from suppliers have decreased $($tab: Defects After Delivery$).$ Upon investigation, she learned that in $2017,$ PLE experienced some quality problems due to an increasing number of defects in materials received from suppliers. The company instituted an initiative in August $2018$ to work with suppliers to reduce these defects, to coordinate deliveries more closely, and to improve materials quality through reengineering supplier production policies. Elizabeth noted the program appeared to reduce an increasing trend in defects; she would like to predict what might have happened had the supplier initiative not been implemented and how the number of defects might be further reduced soon.

$1.$ Consider the defects data from January $2017$ through August $2018.$ Build a scatterplot of the defects over time. Fit $(1)$ a simple linear regression model to the data, $(2)$ a polynomial trendline of order $2,$ and $(3)$ a polynomial trendline of order $3.$ Which model is the best fit for the data?

$2.$ Is the best fit model from question $1$ a good predictor $($at the time$)$ of future defects?

$1$ This case is adapted from Evans, J$.$ R$.(2016).$ Business Analytics: Methods, Models, and Decisions $(2$nd ed$.).$ Upper Saddle River: Pearson

$3.$ Is linear regression an appropriate model to use for making a prediction of defects between January $2017$ and August $2018?$

$4.$ Is the variable in your linear regression model significant?

$5.$ Do the residuals in your linear regression model suggest any potential issues with the model?

$6.$ Now consider all defects data, from January $2017$ through December $2021.$ Build a scatterplot of

the defects over time. Fit a simple linear regression model to the data and a polynomial trendline

of order $2.$ Which model is the best fit for the data?

$7.$ Is the best fit model from question $6$ a good predictor of future defects?

$8.$ When is the first month that PLE realized a clear drop in defects? Enter your answer as a month

number; example: $89.$

$9.$ Did that drop occur immediately after the supplier initiative was implemented in August $2018?$

$10.$ Predict the number of defects for January $2022.$ Enter your solution as a whole number; example:

$42.$

In meeting with PLE$$s human resources director, Elizabeth also discovered a concern about the

high rate of turnover in its field service staff. Senior managers have suggested that the department look closer at its recruiting policies, particularly to try to identify the characteristics of individuals that lead to greater retention. However, in a recent staff meeting, HR managers could not agree on these characteristics. Some argued that year of education and grade point average were good predictors. Others argued that hiring more mature applicants would lead to greater retention. To study these factors, the staff agreed to conduct a statistical study to determine the effect that years of education, college grade point average, and current age have on retention. A sample of $40$ field service engineers was selected to determine the influence