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. state the alternative hypothesis. choose the correct answer below. a. ( h_a \bar{x}=293 ) b. ( h_a mu_x=250 ) c. ( h_a mu_x>250 ) d. ( h_a \bar{x}>293 ) e. ( h_a mu_x>293 )

Explanation:

In hypothesis testing, the alternative hypothesis \(H_A\) is what we are trying to find evidence for. Here, Dora wants to check if the average height of Mountain Ash trees (\(\mu_x\), population mean) is taller than the average height of coastal Douglas - Firs (which is 250 feet). So, the alternative hypothesis should express that the population mean \(\mu_x\) is greater than 250.

• Option A: \(H_A:\bar{x} = 293\) is incorrect because \(\bar{x}\) is the sample mean. Hypotheses are about population parameters (\(\mu\) in this case, not the sample statistic \(\bar{x}\)).
• Option B: \(H_A:\mu_x=250\) is incorrect as it is a statement of equality (which is for the null hypothesis in a one - sided test where we are testing for a difference in a particular direction).
• Option C: \(H_A:\mu_x > 250\) is correct as it is a statement about the population mean (\(\mu_x\)) being greater than the known average height of the other type of tree (250 feet).
• Option D: \(H_A:\bar{x}>293\) is incorrect because it is about the sample mean (\(\bar{x}\)) instead of the population mean (\(\mu_x\)).
• Option E: \(H_A:\mu_x > 293\) is incorrect. The value we are comparing to is the average height of the coastal Douglas - Firs (250 feet), not the sample mean of the Mountain Ash trees (293 feet).

Answer:

C. \(H_A:\mu_x > 250\)