To test the effects of acid rain on the growth of white pine trees, a conservationist measures the diameter (in centimeters) of 200 white pines in the Allegheny National Forest in northwest Pennsylvania.The average diameter of the sampled trees is 23.42 cm.Does the sample provide sufficient evidence to conclude that the current mean diameter of the white pines is less than that recorded 50 years ago (μ = 24.8) if the standard deviation is 6 cm? Calculate the p-value and make an appropriate decision.(Use α = 0.05).
A) p-value = 0.9994; fail to reject H₀
B) p-value = 0.0006; reject H₀
C) p-value = 0.0006; fail to reject H₀
D) p-value = 0.9994; reject H₀

Question

Quizplus · Accepted Answer

We'll use a one-sample t-test to test the hypothesis:
H₀: μ = 24.8 (the current mean diameter of the white pines is 24.8 cm)
H₁: μ < 24.8 (the current mean diameter of the white pines is less than 24.8 cm)
The sample size is large enough to use the t-test, because n = 200 > 30. The test statistic is:

t = (x̄ - μ) / (s / √n) = (23.42 - 24.8) / (6 / √200) = -4.17

where x̄ = 23.42 is the sample mean and s = 6 is the sample standard deviation.

The degrees of freedom are df = n - 1 = 199. Using a t-distribution table or calculator with df = 199, the p-value for a one-tailed test with t = -4.17 is approximately 0.0006.

Since the p-value (0.0006) is less than the significance level α = 0.05, we reject the null hypothesis H₀. We have sufficient evidence to conclude that the current mean diameter of the white pines is less than that recorded 50 years ago.

To Test the Effects of Acid Rain on the Growth