Question

Suppose that you are given a data set which predicts the price of coffee based on loss of forest in Indonesia. Software tells us that the least-squares slope from 10 different observations is b = 0.54 with standard error SE_b = 0.1. Please use 2 decimal places.

What is the t statistic for testing H₀: ? = 0?



How many degrees of freedom does t have?



Use a t-table to approximate the P-value of t against the one-sided alternative H_a: ? > 0. What do you conclude? (select one)

We have no evidence (P > 0.05) that the population slope ? is positive.

We have overwhelming evidence (P < 0.001) that the population slope ? is positive.    

We have slight evidence (0.05 < P < 0.1) that the population slope ? is positive.

We have strong evidence (0.005 < P < 0.01) that the population slope ? is positive.

Answer 1

Null hypothesis, \(\mathrm{H}_{0}: \beta=0\) Alternative hypothesis, \(\mathrm{H}_{a}: \beta>0\) Level of significance, \(\alpha=0.05\) Decision rule: \(\quad\) Reject \(\mathrm{H}_{0}\) if \(\mathrm{P}\) -value \(\leq 0.05\). Summarize the available information:

Slope, \(b=0.54, \quad \mathrm{SE}_{b}=0.10, \quad n=10\)

To test the hypothesis, define the test statistic as follows:

\(t=\frac{b-\beta}{S E_{b}}=\frac{0.54-0.00}{0.10}=5 . \mathbf{4 0}\)

Therefore, the value of \(t\) statistic is \(t=5.40\).

Degrees of freedom, \(\begin{aligned} d f &=n-2 \\ &=10-2 \\ &=8 \end{aligned}\)

Therefore, the statistic \(t\) has 8 degrees of freedom.

For a right-tailed test, compute the P-value as follows:

\(\begin{aligned} \mathrm{P} \text { -value } &=\mathrm{P}(t \geq 5.40) \\ &=\mathrm{TDIST}(5.40,8,1) \\ &=0.000323 \quad[\text { Using Excel function }] \\ & \approx 0.0003 \end{aligned}\)

It is observed that the \(\mathrm{P}\) -value \((0.0003)\) is less than given level of significance \((0.05)\), thus reject \(\mathrm{H}_{0}\)

We have overwhelming evidence \((P<0.001)\) that the population slope \(\beta\) is positive.