Question

shown below is the confidence interval (cl) for the difference, μ₁ - μ₂, between two population means. interpret the confidence interval. 90% cl is from - 30 to 15 choose the correct answer below a. it can be said, with 90% confidence, that the value of μ₁ is somewhere between 15 less and 30 greater than the value of μ₂. note that this means the difference between the two means may be zero. b. the true value of μ₁ - μ₂ lies somewhere between - 30 and 15. note that this means the difference between the two means may be zero. c. it can be said, with 90% confidence, that the value of μ₁ is somewhere between 30 less and 15 greater than the value of μ₂. note that this means the difference between the two means may be zero. d. it can be said, with 90% confidence, that there is no significant difference between the value of μ₁ and the value of μ₂. note that this means the difference between the two means is less than zero.

Explanation:

A confidence interval for \(\mu_1-\mu_2\) gives a range of values within which the true difference between the population means is likely to lie. A 90% confidence interval means that if we were to take many samples and construct confidence intervals in the same way, about 90% of them would contain the true value of \(\mu_1 - \mu_2\).

Option A: Incorrect interpretation of the confidence interval bounds in terms of \(\mu_1\) and \(\mu_2\) relationship.
Option B: Just states the interval without the confidence level interpretation.
Option C: Correctly interprets the confidence interval in terms of \(\mu_1\) and \(\mu_2\) relationship (\(\mu_1-\mu_2\in[- 30,15]\) implies \(\mu_1=\mu_2 + (\mu_1 - \mu_2)\), so \(\mu_1\) is between \(\mu_2-30\) and \(\mu_2 + 15\)) and mentions that the difference may be zero (since \(0\in[-30,15]\)).
Option D: Incorrect, as the confidence interval does not claim "no significant difference" in the way described. The fact that \(0\) is in the interval is just one aspect of the interval's range.

Answer:

C. It can be said, with 90% confidence, that the value of \(\mu_1\) is somewhere between 30 less and 15 greater than the value of \(\mu_2\). Note that this means the difference between the two means may be zero.