If a video game sells well in North America, how well will it sell worldwide?

• Game studios and publishers must decide where to invest marketing money.
• North America is the largest market for most games in this dataset.
• Can early NA numbers give a reliable forecast of global sales?

Statistical idea: Simple linear regression and correlation, with a confidence interval and a hypothesis test.

• Source: https://www.kaggle.com/datasets/gregorut/videogamesales

Interpretation: The pattern is strongly upward, but a few blockbusters (like Wii Sports) stretch the axes and most games are crowded near zero.

Model for game \(i\):

\[y_i = \beta_0 + \beta_1 x_i + \varepsilon_i, \qquad \varepsilon_i \sim N(0,\sigma^2)\]

Least squares chooses \(b_0, b_1\) to minimize the sum of squared residuals:

\[\min_{b_0,b_1} \sum_{i=1}^{n} \left(y_i - b_0 - b_1 x_i\right)^2\]

with solution

\[b_1 = \frac{\sum_{i}(x_i-\bar x)(y_i-\bar y)}{\sum_{i}(x_i-\bar x)^2}, \qquad b_0 = \bar y - b_1 \bar x\]

Meaning: \(b_1\) is the predicted change in global sales (millions) for each extra million sold in North America.

Correlation measures the strength of the linear relationship:

\[r = \frac{\sum_{i}(x_i-\bar x)(y_i-\bar y)}{\sqrt{\sum_{i}(x_i-\bar x)^2\,\sum_{i}(y_i-\bar y)^2}} \qquad R^2 = r^2\]

Hypothesis test for the slope:

\[H_0:\ \beta_1 = 0 \quad \text{vs.} \quad H_a:\ \beta_1 \neq 0, \qquad t = \frac{b_1 - 0}{SE(b_1)} \sim t_{n-2}\]

95% confidence interval for the slope:

\[b_1 \ \pm\ t^{*}_{n-2}\, SE(b_1)\]

# Read and clean the data
vg <- read_csv("vgsales.csv", na = c("", "NA", "N/A"))
vg_na <- vg %>% filter(NA_Sales > 0)

# Fit the simple linear regression
fit <- lm(Global_Sales ~ NA_Sales, data = vg_na)

summary(fit)$coefficients   # estimates, SE, t, p-value

##               Estimate  Std. Error    t value     Pr(>|t|)
## (Intercept) 0.03172333 0.005787590   5.481268 4.307538e-08
## NA_Sales    1.80313216 0.005755948 313.264171 0.000000e+00

confint(fit, level = 0.95)  # 95% CI for b0 and b1

##                  2.5 %     97.5 %
## (Intercept) 0.02037873 0.04306794
## NA_Sales    1.79184958 1.81441473

cor(vg_na$NA_Sales, vg_na$Global_Sales)   # correlation r

## [1] 0.9435349

• Slope \(b_1 =\) 1.803
  • each extra 1 million NA copies is associated with about 1.8 million more copies worldwide.
• 95% CI: (1.792, 1.814)
  • so the slope is estimated precisely.
• Correlation \(r =\) 0.944 and \(R^2 =\) 0.89
  • NA sales explain about 89% of the variation in global sales.
• Test: \(t =\) 313.3 and p-value < 2.2e-16 → strong evidence to reject \(H_0:\beta_1=0\).

Interpretation: The shaded band is the uncertainty in the average global sales at each NA value. It is narrow, so the trend is very reliable.

Interpretation: The residuals fan out as sales grow (non-constant variance) and a few blockbusters are large positive outliers. The slope is trustworthy as a trend, but individual predictions for hits are far less certain than the band suggests.

Games with at least 0.5M global sales. Global sales rise with both NA and EU sales, which hints that a multiple regression using both would predict even better.

• Strong, positive linear relationship between NA sales and Global sales
  • (\(r \approx\) 0.94, \(R^2 \approx\) 0.89.)
• Each extra million NA copies goes with about 1.8 million more copies globally
  • 95% CI: 1.79 to 1.81
• The slope is statistically significant
  • p-value is essentially 0
• Possibly looking at the both NA sales and EU sales will be a more accurate predictor of Global sales.