Claire H.

asked • 04/17/17

confident intervals and central limit theorem.

A random sample of 39 cows was fed a special high-protein grain for a month. Each was weighed before and after and the average weight gained was 6.7 pounds, with standard deviation 7.1.
i. Construct and interpret a 99% confidence for the true mean weight gain.
ii. Briefly (i.e. in one-two sentences) explain how the Central Limit Theorem is used in part i. 

1 Expert Answer

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Jason L. answered • 04/17/17

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1) A 99% confidence interval is +/- 2.575 standard deviations from the mean. So the formula for the interval is:
 
6.7 +/- (2.57)(7.1)
 
2) The CLT states that as your sample size increases, the mean of your sample will approach the true mean of the entire population. So how would that apply to #1? A 99% confidence interval gives a very wide range for your population weight gain. What do you think would happen to your confidence interval if they had more cows to weigh?
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Claire H.

1). I understand where you got 2.57 I got that myself but my problem lies after that... is the formula for the confidence interval:  mean +/- (Standard Error)(Zscore) or is it as you said above and it is the standard deviation instead of standard error?? In my notes I have it as the standard error and I get an interval of [-2.55, 15.95], the fact that I have a negative number confuses me but I guess that this could also mean weight loss? But then again do you think I have the wrong formula? 
 
2.) Wold my confidence interval become within a smaller range, hence more representative of the true mean? 
 
Thanks for your time Jason, I appreciate it. 
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04/18/17

Jason L.

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Okay I see why my answer was off. Yes, the formula is standard error of the mean * z-score +/- sample mean. You'd use standard error of the mean when measuring the sample from a population.
 
So using the correct formula, we should get:
 
6.7 +/- (7.1 / √39) * 2.575
[3.77, 9.62]
 
How did you get the interval you got? What steps did you take? 
 
PS - A negative number would still have made sense because it would just mean the average cow's weight within the sample actually decreased instead of increased.
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04/18/17

Claire H.

Sorry I put the n as 3.9 into my calculator instead of 39. I am now getting the same interval as you. Thanks for all your time and help! 
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04/20/17

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