You have collected data for the 50 U.S.states and estimated the following relationship between the change in the unemployment rate from the previous year ( )and the growth rate of the respective state real GDP (gy).The results are as follows = 2.81 - 0.23 gy,R2= 0.36,SER = 0.78 (0.12)(0.04)
Assuming that the estimator has a normal distribution,the 95% confidence interval for the slope is approximately the interval
A)[2.57,3.05]
B)[-0.31,0.15]
C)[-0.31,-0.15]
D)[-0.33,-0.13]

Question

You have collected data for the 50 U.S.states and estimated the following relationship between the change in the unemployment rate from the previous year (  )and the growth rate of the respective state real GDP (gy).The results are as follows  = 2.81 - 0.23  gy,R2= 0.36,SER = 0.78 (0.12)(0.04)
Assuming that the estimator has a normal distribution,the 95% confidence interval for the slope is approximately the interval
A)[2.57,3.05] 
B)[-0.31,0.15] 
C)[-0.31,-0.15] 
D)[-0.33,-0.13]

Quizplus · Accepted Answer

The 95% confidence interval for the slope is given by:
b1 ± t(α/2, df) × SE(b1)
where b1 is the estimated coefficient of gy, t(α/2, df) is the t-value for a two-tailed test with α/2 level of significance and df degrees of freedom (df = n-2 for simple linear regression), and SE(b1) is the standard error of the estimator.

In this case, we have b1 = -0.23, SE(b1) = 0.04, df = 50-2 = 48, and t(0.025, 48) = 2.01 (from t-distribution table or calculator). Substituting these values, we get:

-0.23 ± 2.01 × 0.04
= -0.23 ± 0.08
= [-0.31, -0.15]

Therefore, the correct answer is C, which is the interval [-0.31, -0.15]. This means that we are 95% confident that the true population slope lies somewhere in this interval. Since the interval does not include zero, we can conclude that there is a significant negative relationship between the change in unemployment rate and the growth rate of real GDP, i.e., as real GDP grows, unemployment rate tends to decrease.

You Have Collected Data for the 50 U