Part II. A study of the effects of acid rain on trees in the Hopkins Forest, a w
ID: 3352567 • Letter: P
Question
Part II. A study of the effects of acid rain on trees in the Hopkins Forest, a woodland reserve covering parts of Massachusetts, New York and Vermont, shows that 25 out of 100 trees sampled exhibited some sort of damage from acid rain. However, a recent article in the journal Environmetrics reports that 15% of trees in the Northeast, on average, exhibit damage from acid rain. A. ONE-SIDED or TWO-SIDED (circle one). Researchers want to test the hypothesis that the trees in the Hopkins Forest are more susceptible to the effects of acid rain than trees in the rest of the Northeastern US. (Score: 1 or 0) B. The researchers failed to reject the null hypothesis for this test of statistical significance (a .05). What do the researchers conclude? (Score: 1 or 0) Researchers decide to compare the damage from acid rain found in the two forests. Twenty-five out of 100 randomly selected trees in the Hopkins Forest exhibited damage; 20 out of 108 randomly selected trees in The White Mountain National Forest exhibited damage. C. Write the hypotheses that would be used to determine whether there is a difference in acid rain damage observed at the two forests. (Score: 1 or 0) D. If the researchers had set = .01, keeping all else the same, they: (Score: 1 or 0) Would definitely have found evidence of a true difference. Might have found evidence of a true difference. Would not have found evidence of a true difference. a. b. c.Explanation / Answer
Solution:-
A) One-sided
State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.
Null hypothesis: P < 0.15
Alternative hypothesis: P > 0.15
Note that these hypotheses constitute a one-tailed test.
B) The researcher concludes that the trees in the Hopkins forest are not more susceptible to the the effects of acid rain than trees in the rest of the Northeastern US.
C)
State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.
Null hypothesis: P1 = P2
Alternative hypothesis: P1 P2
Note that these hypotheses constitute a two-tailed test.
D)
State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.
Null hypothesis: P1 = P2
Alternative hypothesis: P1 P2
Note that these hypotheses constitute a two-tailed test.
Formulate an analysis plan. For this analysis, the significance level is 0.01. The test method is a two-proportion z-test.
Analyze sample data. Using sample data, we calculate the pooled sample proportion (p) and the standard error (SE). Using those measures, we compute the z-score test statistic (z).
p = (p1 * n1 + p2 * n2) / (n1 + n2)
p = 0.21635
SE = sqrt{ p * ( 1 - p ) * [ (1/n1) + (1/n2) ] }
SE = 0.05714
z = (p1 - p2) / SE
z = 1.13
where p1 is the sample proportion in sample 1, where p2 is the sample proportion in sample 2, n1 is the size of sample 1, and n2 is the size of sample 2.
Since we have a two-tailed test, the P-value is the probability that the z-score is less than - 1.13 or greater than 1.13.
We use the Normal Distribution Calculator to find P(z < -1.13) and P(z > 1.13) = 0.2628
Interpret results. Since the P-value (0.2628) is greater than the significance level (0.01), we have to accept the null hypothesis.
C) Would not have found evidence of a true difference.