Maths

👨🏻‍🏫 學術台數學

[匿名]

2011-09-07 5:43 pm

Given :
x + 0.1645 sqrt [x (1 - x)] = 0.455 ...........(1) and
x - 0.1645 sqrt [x(1 - x)] = 0.295...............(2)
(1) + (2), 2x = 0.455 + 0.295, x = 0.375.
(1) - (2), 2(0.1645)sqrt[x(1- x)] = 0.16
Solving the equation, x = 0.61615 or 0.38385, which is not 0.375!
Which one is correct?
Where is the mistake?

更新1:

To : 學問 Do you mean all 3 of them are roots to the 2 equations? But when substitute 0.61615 or 0.38385 into the equations, they are wrong!!, why? Please explain further, thanks.

更新2:

To : 自由自在. Thank you for your explanation. This question is in fact asking to find the sample proportion with confidence interval (0.295, 0.455) at confidence level 90% and sample size 100, that is (p - 1.645 sqrt[p(1-p)/100], p + 1.645 sqrt[p(1-p)/100]).

更新3:

To : 自由自在. Obviously, the simplest way is using (1) + (2), but why (2) - (1) doesn't work? why solving individual equations doesn't work? Are the 2 equations really inconsistent?

回答 (2)

用戶9112

2011-09-08 6:13 am

✔ 最佳答案

http://i1090.photobucket.com/albums/i376/Nelson_Yu/int-18.jpg

圖片參考：http://i1090.photobucket.com/albums/i376/Nelson_Yu/int-18.jpg

2011-09-08 20:35:15 補充：
(1) 0.1645 is not an exact value. A more accurate value should be 0.164485362695147
(2) Using normal distribution to estimate binomial distribution is only approximation only.
Therefore all the numbers in your original equations are not exact, but approximate values only.

2011-09-08 20:38:21 補充：
This explains why the 2 equations are not consistent.
But if you interpret the question to mean to find the approximate value of p if the 90% confidence interval is approximately (0.295, 0.455) then we can solve the equation using all means:

2011-09-08 20:38:27 補充：
Method 1. Solving individual equations
Method 2. (1) + (2)
Method 3. (1) - (2)
Method 4. (1) * (2)
All are valid methods.

2011-09-08 20:40:55 補充：
But each method cannot give exact result, only approximate results since the 2 equations are approximate equations. But each method will give similar result of varied accuracy. Indeed Method 1, 2 and 4 all yield similar result close to 0.375 and method 3 gives 0.38385 with great error tolerance.

2011-09-08 20:46:32 補充：
To satisfy yourself with this explanation, you can try to solve the same equations by your 2 methods, but change 0.295 to 0.2954 and 0.455 to 0.4546, then the result will becomes closer. Try again 0.29536 and 0.45464, and you will get still closer result.

2011-09-08 20:46:41 補充：
I mean here that both methods are correct, only that each method gives result of different accuracy,

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學問

2011-09-08 6:26 pm

自由自在兄's answer is much better~

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