Understanding Confidence Intervals. Please comment.
Bacle
am trying to answer.( I will mark my answers with a ------- to make it easier to recognize. Please feel free to check just one-or-two of the answers if this seems too long). I would appreciate your comments:
Here is the problem:
Teachers
Software analysis of the salaries of a random sample of 288 Nevada teachers produced the confidence interval shown below. Which conclusion is correct? What's wrong with the others?
t-Interval for m: with 90.00% Confidence, 38944 < m(TchPay) < 42893
a)If we took many random samples of Nevada teachers, about 9 out of 10 of them would produce this confidence interval.
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a)False: the confidence interval would depend on the value of sampling mean. Since we are using t-intervals, we must be using the sample error, which makes intervals even more variable than if we knew the true pop. standard deviation.
All we can say is that there is a 95% probability that
the true average salary lies in a 95% confidence interval, whatever interval we construct.
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b)If we took many random samples of Nevada teachers, about 9 out of 10 of them would produce a confidence interval that contained the mean salary of all Nevada teachers.
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True, if we constructed 95% confidence t-intervals with the sampling data given.
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c)About 9 out of 10 Nevada teachers earn between $38,944 and $42,893.
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False. The confidence interval is about the true population mean, about the probability that the true mean lies in the interval, not about the probability that a teacher earns an amount in this range.
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d)About 9 out of 10 of the teachers surveyed earn between $38,944 and $42,893.
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False.
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d)We are 90% confident that the average teacher salary in the United States is between $38,944 and $42,893.
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d)True. This is the actual meaning of a confidence interval.
Thanks For Any Comments.
Rich Ulrich
>Could someone please tell me if I am understanding confidence intervals correctly.?. Here is a problem I
>am trying to answer.( I will mark my answers with a ------- to make it easier to recognize. Please feel free to check just one-or-two of the answers if this seems too long). I would appreciate your comments:
>
>
>Here is the problem:
>
> Teachers
>Software analysis of the salaries of a random sample of 288 Nevada teachers produced the confidence interval shown below. Which conclusion is correct? What's wrong with the others?
>
>t-Interval for m: with 90.00% Confidence, 38944 < m(TchPay) < 42893
>
>
>a)If we took many random samples of Nevada teachers, about 9 out of 10 of them would produce this confidence interval.
That is not a "conclusion." That is an erroneous statement of
a definition.
>
>---------------------------------------------------
>a)False: the confidence interval would depend on the value of sampling mean. Since we are using t-intervals, we must be using the sample error, which makes intervals even more variable than if we knew the true pop. standard deviation.
>
> All we can say is that there is a 95% probability that
> the true average salary lies in a 95% confidence interval, whatever interval we construct.
Not too bad. Except that the original was "90%" and not 95%.
>________________________________________________
>
>
>b)If we took many random samples of Nevada teachers, about 9 out of 10 of them would produce a confidence interval that contained the mean salary of all Nevada teachers.
Again, that is a definition, not a "conclusion."
>
>-----------------------------------------------
> True, if we constructed 95% confidence t-intervals with the sampling data given.
Again, 95 is not 90.
>____________________________________________________
>
>c)About 9 out of 10 Nevada teachers earn between $38,944 and $42,893.
>
>------------------------------------------------------
>
>False. The confidence interval is about the true population mean, about the probability that the true mean lies in the interval, not about the probability that a teacher earns an amount in this range.
>
>_________________________________________________________
>
>d)About 9 out of 10 of the teachers surveyed earn between $38,944 and $42,893.
>------------------------------------------------------
>
>False.
>_____________________________________________________
>
>d)We are 90% confident that the average teacher salary in the United States is between $38,944 and $42,893.
We don't freely extrapolate from "Nevada" to "the United States."
>
>-------------------------------------------------
>
>d)True. This is the actual meaning of a confidence interval.
The definition that you gave in response to (a) is a better
defintion. "90% confident" is an awkward expression
whose meaning would never be guessed by someone who
doesn't know the conventional answer.
--
Rich Ulrich