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, what is the conclusion?
choose the correct answer below.
a. there is not sufficient evidence to indicate that the average height of all mountain ash trees in southeast australia is more than 250 feet.
b. there is sufficient evidence to indicate that the average height of all mountain ash trees in southeast australia is equal to 250 feet.
c. there is sufficient evidence to indicate that the average height of the mountain ash trees in this sample is more than 250 feet.
d. there is sufficient evidence to indicate that the average height of all mountain ash trees in southeast australia is more than 250 feet.
e. there is not sufficient evidence to indicate that the average height of the mountain ash trees in this sample is more than 250 feet.

Explanation:

Step1: Analyze the t - statistic

A large t - statistic (in this case \(t = 6.661\)) indicates a significant difference between the sample mean and the hypothesized population mean.

Step2: Consider the hypothesis

The null hypothesis \(H_0:\mu=250\) (where \(\mu\) is the population mean height of Mountain Ash trees) and the alternative hypothesis \(H_1:\mu > 250\). A large positive t - statistic provides evidence against \(H_0\) in favor of \(H_1\).

Step3: Evaluate the options

• Option A: Incorrect, because the large t - statistic provides evidence.
• Option B: Incorrect, the t - statistic is for a one - sided test (\(\mu>250\)), not for equality.
• Option C: Incorrect, the sample mean is already known (\(\bar{x} = 293\)), the test is about the population mean.
• Option D: Correct, the large t - statistic provides sufficient evidence to reject \(H_0\) in favor of \(H_1\) (that the population mean \(\mu>250\)).
• Option E: Incorrect, the sample mean is \(293>250\) and the t - statistic is for the population mean.

Answer:

D. There is sufficient evidence to indicate that the average height of all Mountain Ash trees in southeast Australia is more than 250 feet.