Question

dora likes to explore. recently, she explored southeast australia where she found some very large trees known as eucalyptus regnans or mountain ash. she wondered if they were taller, on average, than the coastal douglas firs of her native state of oregon in the united states, which have an average height of 250 feet in old growth areas. dora measured the heights of 15 mountain ash trees in southeast australia and found an average height of these trees of 293 feet. suppose s = 25 feet. assume the heights of the 15 trees in doras sample are representative of the heights of all mountain ash trees in southeast australia. the t - statistic for this problem is 6.661. based on this t - statistic, which of the following is true? choose the correct answer below. a. with a p - value less than 0.0001, there is sufficient evidence to reject the null hypothesis and accept the alternative as true. b. with a p - value of 0.999, there is sufficient evidence to accept the null hypothesis as true. c. with a p - value less than 0.0001, there is not sufficient evidence to accept the null hypothesis as true. d. with a p - value less than 0.0001, there is not sufficient evidence to reject the null hypothesis and accept the alternative as true. e. with a p - value of 0.999, there is not sufficient evidence to reject the null hypothesis and accept the alternative as true.

Explanation:

Step1: Determine the degrees of freedom

The degrees of freedom \(df=n - 1\), where \(n = 15\). So \(df=15-1 = 14\).

Step2: Analyze the t - statistic and p - value relationship

For a one - sided t - test with \(df = 14\), when \(t=6.661\). Using a t - table or statistical software, we know that for a t - distribution with \(df = 14\), a very large t - value (in absolute value for a two - sided test, and positive for a right - sided test as we expect if we are testing if Mountain Ash is taller) corresponds to a very small p - value.
Since \(t = 6.661\) is extremely large, the p - value is less than \(0.0001\).
When the p - value is less than the significance level (commonly \(\alpha=0.05\) or \(\alpha = 0.01\)), we reject the null hypothesis.

Answer:

A. With a p - value less than \(0.0001\), there is sufficient evidence to reject the null hypothesis and accept the alternative as true.