I think this is about the Central Limit Theorem

[whitejac]

Homework Statement

An engineer is measuring a quantity q. It is assumed that there is a random error in each measurement, so the engineer will take n measurements and reports the average of the measurements as the estimated value of q. Specifically, if Y_i is the value that is obtained in the i'th measurement, we assume that

Y_i=q+X_i,
where X_i is the error in the ith measurement. We assume that Xi's are i.i.d. with EX_i=0 and Var(X_i)=4 units. The engineer reports the average of measurements

M_n = (Y₁ + ... Y_n) / n

How
many measurements does the engineer need to make until he is 95% sure that the final error is less than 0.1 units? In other words, what should the value of n be such that
P(q−0.1≤M_n≤q+0.1)≥0.95?

Homework Equations

Central Limit Theorem states:
Z_n = (M_x - μ) / (σ / √n)

The Attempt at a Solution

So this is the formula I chose to use. It seems like a simple variable swap, but my problem is pulling n out.

P(y₁ ≤ Y ≤ y₂)
= P( (y₁ - nμ) / (σ√n) ≤ ((X₁ +... X_n) - nμ) / (σ√n)) ≤ (y₂ - nμ) / (σ√n)

Then this would give ((q + 0.1) / (2√n)) - ((q - 0.1) / (2√n)) = Φ^-1 (0.95) = 1.64

combining this would give:(q/2√n) + (0.1 / 2√n) - (q / 2√n)- (0.1 / 2√n) = 0.2 / 2√n = 1/√n = 1.64 ⇒ n = (1/1.64)² = 0.37

Now... this is not an integer, and makes absolutely no sense. I am semi-confident in my process, but I think I may have made too many assumptions about the central limit theorem.

 

[BvU]

Why do you change from $\ \sigma\over\sqrt n\$ to $\ \sigma\sqrt n\$ ?

It seems your n is used for two different purposes as well

...

 

[whitejac]

Why do you change from $\ \sigma\over\sqrt n\$ to $\ \sigma\sqrt n\$ ?

It seems your n is used for two different purposes as well

...

When you multiply by n/n you get σ√n

 

[BvU]

I would call that multiplying by n, not multiplying by n/n

P(y1 ≤ Y ≤ y2)
= P( (y1 - nμ) / (σ√n) ≤ ((X1 +... Xn) - nμ) / (σ√n)) ≤ (y2 - nμ) / (σ√n)

Then this would give ((q + 0.1) / (2√n)) - ((q - 0.1) / (2√n)) = Φ-1 (0.95) = 1.64

Check it anyway !

 

[whitejac]

Central Limit Theorem states:
Zn = (Mx - μ) / (σ / √n)

It has to be n/n because Mx is (X1 +...+Xn)/n.

In my extrapolation of the formula you quoted, the y1 = q - 0.1, nμ = 0 and σ = √4
Which portion did I misinterpret/use twice?

 

[BvU]

The "Then" part, where you divide by n

Another comment: Your 1.64 looks one-sided to me. And your notation asks more from the reader than is reasonable; teachers shouldn't allow that ! (there is no using n twice; I got confused -- sorry)

I end up with an awful lot of measurements required, which is rather to be expected if you start out with such a large variance !

By the way, even "Var(Xi)=4 units" is confusing. Shouldn't it be Var(Xi)=4 units² ?More (admittedly nitpicking)
Central Limit Theorem states: Z_n = (M_x - μ) / (σ / √n) is normally distributed with standard deviation 1

Matter of completeness: for some readers Z_n is something else: an impedance or whatever.

 

[whitejac]

By the way, even "Var(Xi)=4 units" is confusing. Shouldn't it be Var(Xi)=4 units2 ?

Sorry, it ought to be. My book isn't the best despite my professor's claim. I guess they're assuming "units" is general enough.

The "Then" part, where you divide by n

Maybe I'm not looking at it the same way you are, I don't divide by n. I multiply by n to get y1 and y2 = q-0.1, q+0.1, respectively. This is consistent with examples in my book, I think. Are you saying I mistook these values as y1,y2 when they were in fact y1 - nμ, y2 + nμ?

I apologize for my not showing as many steps, or in the best way. It's much neater on my paper. I'll upload it when i get home if it remains unclear.

 

[BvU]

No need to apologize, I'm trying to help.

P(y1 ≤ Y ≤ y2)
= P( (y1 - nμ) / (σ√n) ≤ ((X1 +... Xn) - nμ) / (σ√n)) ≤ (y2 - nμ) / (σ√n)

Then this would give ((q + 0.1) / (2√n)) - ((q - 0.1) / (2√n)) = Φ-1 (0.95) = 1.64

In the top line you have y₁ for nq, in the bottom line you have q only. You only divide the numerator by n.

The 1.64 is for 5% in one tail, you want 2.5% in each tail.

--

[edit] The more I read it, the more nonsensical it becomes:

P(y1 ≤ Y ≤ y2) = P( (y1 - nμ) / (σ√n) ≤ ((X1 +... Xn) - nμ) / (σ√n)) ≤ (y2 - nμ) / (σ√n)

What does it say here ?

 

[Ray Vickson]

Sorry, it ought to be. My book isn't the best despite my professor's claim. I guess they're assuming "units" is general enough.

Maybe I'm not looking at it the same way you are, I don't divide by n. I multiply by n to get y1 and y2 = q-0.1, q+0.1, respectively. This is consistent with examples in my book, I think. Are you saying I mistook these values as y1,y2 when they were in fact y1 - nμ, y2 + nμ?

I apologize for my not showing as many steps, or in the best way. It's much neater on my paper. I'll upload it when i get home if it remains unclear.

I suggest you go back to square one. You need to figure out how to make a 95% probability interval (centered at the mean) come out with a length of 0.2 (i.e., from -0.1 to + 0.1). In terms of the unit normal distribution, how many standard deviations wide would that be? Then carefully figure out what that means in terms of n. When I do it I get a value of n between 1000 and 2000, but I will not state the precise value.