Question

An article used a multiple regression model with four independent variables to study accuracy in reading liquid crystal displays. The variables were

y =	error percentage for subjects reading a four-digit liquid crystal display
x1 =	level of backlight (ranging from 0 to 122 cd/m2)
x2 =	character subtense (ranging from 0.025° to 1.34°)
x3 =	viewing angle (ranging from 0° to 60°)
x4 =	level of ambient light (ranging from 20 to 1500 lux)

The model fit to data was Y = β₀ + β₁x₁ + β₂x₂ + β₃x₃ + β₄x₄ + ε. The resulting estimated coefficients were ₀ = 1.52, ₁ = 0.01, ₂ = −1.40, ₃ = 0.02, ₄ = −0.0009.

(a) The estimated model was based on n = 30 observations, with SST = 39.5 and SSE = 22. Calculate the coefficient of multiple determination. (Round your answer to three decimal places.)

(b)Calculate the test statistic using α = 0.05. (Round your answer to two decimal places.)

Answer 1

a) SST = 39.5 and SSE = 22, where SST is total sum of square and SSE is Residual sum of squares

SSR = SST - SSE = 39.5 - 22 = 17.5 where SSR is Regression sum of squares

Coefficient of determination = R² = SSR / SST = 17.5 / 39.5 = 0.443

R² tells about the proportion of variance explained by the model, so 44.3% of the variability is explained by the model

b) There are 5 regressors including a constant so p = 5. Also n = 30

Mean Square due to regression (MSR) = SSR / (p-1) = 17.5 / 4 = 4.375

Mean Square error (MSE) = SSE / (n-p) = 22 / (30-5) = 22 / 25 = 0.88

So, test statistic is F = MSR / MSE = 4.375 / 0.88 = 4.97

So the calculated value is 4.97 and tabulated value is F_4,25,0.05 = 2.759

Since, calculated F > tabulated F , so we reject the null hypothesis and conclude that there is a effect of regressor on Y

Also the p value is P(F > 4.97) = 0.004347734 , so again we can see than p_value < α = 0.05 , so we reject the null hypothesis