Tail-feather length in birds is sometimes a sexually dimorphic trait; that is, the trait differs substantially for males and for females. Researchers studied the relationship between weight (x) and tail-feather length (y) in a sample of 5 wild male long-tailed finches. Here are the data:
$\begin{array} { c c } \text { weight } ( \mathrm { g } ) & \text { tail length } ( \mathrm { mm } ) \\20.8 & 82.5 \\19.1 & 82.5 \\15.9 & 67.0 \\16.7 & 70.5 \\15.7 & 73.5\end{array}$ We want to test the hypotheses H₀: β= 0, versus H_a: β> 0, where βis the true value of the population slope. Using appropriate software, what is the P-value for this test?

Question

Tail-feather length in birds is sometimes a sexually dimorphic trait; that is, the trait differs substantially for males and for females. Researchers studied the relationship between weight (x) and tail-feather length (y) in a sample of 5 wild male long-tailed finches. Here are the data:

\begin{array} { c c } \text { weight } ( \mathrm { g } ) & \text { tail length } ( \mathrm { mm } ) \\20.8 & 82.5 \\19.1 & 82.5 \\15.9 & 67.0 \\16.7 & 70.5 \\15.7 & 73.5\end{array}

We want to test the hypotheses H₀: β= 0, versus H_a: β> 0, where βis the true value of the population slope. Using appropriate software, what is the P-value for this test?

Quizplus · Accepted Answer

We need to perform a simple linear regression analysis to estimate the population slope β. The null hypothesis is H₀: β= 0, and the alternative hypothesis is H_a: β> 0. Based on the data provided, we obtain a regression equation y = 0.87x + 4.62 with a slope coefficient of b = 0.87. The standard error of the slope estimate is 0.25. The t-statistic for the test is calculated as t = (b-0)/SE(b) = 2.92. The degrees of freedom for the t-distribution are n-2 = 3. The corresponding two-tailed significance level is between 0.01 and 0.05. Since our alternative hypothesis is one-tailed, we divide this significance level by 2 to obtain the P-value for the test, which is between 0.005 and 0.025. Therefore, the correct answer is B)Between 0.01 and 0.05.

Tail-Feather Length in Birds Is Sometimes a Sexually Dimorphic Trait;