A researcher analyzes the factors that may influence amusement park attendance.She estimates the following model: Attendance = β₀ + β₁ Price + β₂ Temperature + β₃ Rides + ε,whereAttendance is the daily attendance (in 1,000s),Price is the gate price (in $),Temperature is the average daily temperature (in ^oF),and Rides is the number of rides at the amusement park.A portion of the regression results is shown in the accompanying table. When testing whether the explanatory variables Temperature and Rides are jointly significant,the error sum of squares for the restricted model is SSE_R= 12,343.78.Which of the following is the value of the test statistic when conducting this test? A) −4.09 B) 25.33 C) 49.58 D) 99.78

Question

Quizplus · Accepted Answer

To conduct a joint test of significance for Temperature and Rides, we need to compare the error sum of squares from the restricted model (where Temperature and Rides are excluded) to the error sum of squares from the full model (where Temperature and Rides are included). The test statistic is calculated as: (SSE(restricted) - SSE(full))/(k- p), where k is the number of regressors (including the intercept) in the full model and p is the number of regressors in the restricted model (excluding Temperature and Rides). In this case, k = 5 and p = 3, so the degrees of freedom for the test are df = 2.

Substituting the given values, we get: 
Test statistic = (SSE(restricted) - SSE(full))/(k- p) 
= (12343.78 - 8709.47)/(5-3) 
= 1867.31/2 
= 933.655

Using an F-distribution with numerator degrees of freedom = 2 and denominator degrees of freedom = 341, we find that the critical value at the 5% level of significance is 3.01. Since the test statistic (933.655) is much larger than the critical value (3.01), we reject the null hypothesis that Temperature and Rides are jointly insignificant. Therefore, we can conclude that at least one of these variables is useful in explaining the variation in amusement park attendance. The correct answer is therefore C.

A Researcher Analyzes the Factors That May Influence Amusement Park