The Meaning of a 95% T Confidence Interval for the Mean

[FredericChopin]

Homework Statement

A diet pill is given to 9 subjects over six weeks. The average difference in weight (follow up - baseline) is -2 pounds. What would the standard deviation have to be for the 95% T confidence interval to lie entirely below 0?

ANSWER: Around 2.6 pounds or less

Refer to the previous question. The interval would up being [-3.5, -0.5] pounds. What can be said about the population mean weight loss at 95% confidence?

A: We can not rule out the possibility of no mean weight loss at 95% confidence.

B: There is support of mean weight gain at 95% confidence.

C: There is support at 95% confidence of mean weight loss.

D: We can not rule out the possibility of mean weight gain at 95% confidence.

Homework Equations

The Attempt at a Solution

I have three attempts to answer this question.

On my first attempt, I said D, thinking that since 5% of intervals do not contain the population mean, there is a chance that the population mean could be positive (and there a mean weight gain). But I got the wrong answer, and I don't understand why.

On my second attempt, I thought that "weight gain" wasn't necessarily going to occur but "no mean weight loss" was, so I chose A, and I still got the wrong answer.

I am surprised that confidence intervals will guarantee a support (as the remaining answers of the question suggest), so I now think the answer is C. B sounds ridiculous, because it doesn't make sense why there is support for mean weight gain at 95% confidence.

Can someone help me understanding the question? I hope I finally get this question right.

Thank you.

 

[Ray Vickson]

Homework Statement

A diet pill is given to 9 subjects over six weeks. The average difference in weight (follow up - baseline) is -2 pounds. What would the standard deviation have to be for the 95% T confidence interval to lie entirely below 0?

ANSWER: Around 2.6 pounds or less

Refer to the previous question. The interval would up being [-3.5, -0.5] pounds. What can be said about the population mean weight loss at 95% confidence?

A: We can not rule out the possibility of no mean weight loss at 95% confidence.

B: There is support of mean weight gain at 95% confidence.

C: There is support at 95% confidence of mean weight loss.

D: We can not rule out the possibility of mean weight gain at 95% confidence.

Homework Equations

The Attempt at a Solution

I have three attempts to answer this question.

On my first attempt, I said D, thinking that since 5% of intervals do not contain the population mean, there is a chance that the population mean could be positive (and there a mean weight gain). But I got the wrong answer, and I don't understand why.

On my second attempt, I thought that "weight gain" wasn't necessarily going to occur but "no mean weight loss" was, so I chose A, and I still got the wrong answer.

I am surprised that confidence intervals will guarantee a support (as the remaining answers of the question suggest), so I now think the answer is C. B sounds ridiculous, because it doesn't make sense why there is support for mean weight gain at 95% confidence.

Can someone help me understanding the question? I hope I finally get this question right.

Thank you.

For D: it said "We can not rule out the possibility of mean weight gain at 95% confidence". Of course we cannot rule out the possibility of mean gain altogether, but with 95% confidence, we can. In other words, 95% confidence is not 100% confidence.

Look at your other answers in the same way.

 

[FredericChopin]

For D: it said "We can not rule out the possibility of mean weight gain at 95% confidence". Of course we cannot rule out the possibility of mean gain altogether, but with 95% confidence, we can. In other words, 95% confidence is not 100% confidence.

Look at your other answers in the same way.

I see.

So (theoretically), a 100% confidence interval would be the interval (-∞, +∞). But in this case, a 95% confidence interval is [-3.5, -0.5]. As you said, the important thing is that the question asks what can be said at 95% confidence.

Since the interval does not contain any positive numbers, at 95% confidence, there is no possibility that there can be a mean weight gain (and for that matter, no mean weight loss), so D and A must be incorrect.

Using the same positive-numbers argument, presumably, there is no support for a mean weight gain at 95% confidence, so B must also be incorrect. Therefore, C must be correct.

How is that?

What puzzles me still is why any confidence interval can guarantee a support within that interval. Isn't there the slightest chance that the function is 0 within the interval?

Thank you.

 

[Ray Vickson]

I see.

So (theoretically), a 100% confidence interval would be the interval (-∞, +∞). But in this case, a 95% confidence interval is [-3.5, -0.5]. As you said, the important thing is that the question asks what can be said at 95% confidence.

Since the interval does not contain any positive numbers, at 95% confidence, there is no possibility that there can be a mean weight gain (and for that matter, no mean weight loss), so D and A must be incorrect.

Using the same positive-numbers argument, presumably, there is no support for a mean weight gain at 95% confidence, so B must also be incorrect. Therefore, C must be correct.

How is that?

What puzzles me still is why any confidence interval can guarantee a support within that interval. Isn't there the slightest chance that the function is 0 within the interval?

Thank you.

(1) A 100% confidence interval need not be the whole line; it is just an interval we are SURE contains the true mean---not just "almost sure", but absolutely sure. It could be a point (which it would be in the limit of an infinite sample size, example).
(2) I think you have identified the true/false answers correctly.
(3) Your statement "why any confidence interval can guarantee a support within that interval" is false: there is no such guarantee. We can only be more-or-less sure, but not absolutely, 100% sure. Every once in a while we will be wrong (about 5% of the time if we are looking at a 95% confidence interval).

 

[FredericChopin]

(1) A 100% confidence interval need not be the whole line; it is just an interval we are SURE contains the true mean---not just "almost sure", but absolutely sure. It could be a point (which it would be in the limit of an infinite sample size, example).
(2) I think you have identified the true/false answers correctly.
(3) Your statement "why any confidence interval can guarantee a support within that interval" is false: there is no such guarantee. We can only be more-or-less sure, but not absolutely, 100% sure. Every once in a while we will be wrong (about 5% of the time if we are looking at a 95% confidence interval).

That makes sense. Thank you very much. I did get the right answer, by the way. :)