For the data and sample regression equation shown below, do the following. a. De
ID: 3266918 • Letter: F
Question
Explanation / Answer
3. Assume beta1 denote the slope of the population regression line that relates y to x. The null hypothesis states that x is not an useful predictor for determining versus alternative hypothesis is that x is an useful predictor for y.
H0:beta1=0 Ha:beta1=/=0 Option B. Option A is wrong in sense that null hypothesis states the tentative proposition of the researcher, which shoyl dbe stated as alternative hypothesis. It is asked to determine whether x is useful for predicting y, therefore, sign of the slope should not be taken into account. Option C and D are discarded.
t=b1/(se/sqrt Sxx), where, b1 is coefficient of slope beta1, se is the standard error of the estimate, and Sxx is computed as Sxx=sigma x^2-(sigma x)^2/n, where, n is number of pairs.
Therefore, Sxx=30-(10)^2/4=5
Substitute the given values in the formula to compute t test statistic.
=-1.9/(4.3989/sqrt 5)
=-0.966
The critical t value at alpha/2 (alpha=0.05) and n-2=2 degrees of freedom is: +-4.303.
Per rejection rule based on critical value, reject null hypothesis of observed t>critical t. Here, test statistic (-0.966) does not fall in critical region. Therefore, fail to reject null hypothesis. Thus, there is insuffciient sample evidence to ocnclude that x is an useful predictor of y.
Note Option B and D are outright discarded per rejection rule. Within option A and C, when one fails to reject H0, there is obvioulsy insuffciient sample evidence to support the alternative hypothesis. Option A is correct.
b. The 95% confidence innterval for beta1 is as follows:
b1+-talpha/2. (se/sqrt Sxx)
=-1.9+-4.303(4.3989/sqrt 5)
=(-10.365, 6.565) [ans]
1. talpha/2=+-4.303 (alpha=0.05, alpha/2=0.025, n=4, n-2=2 degrees of freedom)
-talpha=-4.303
talpha=4.303