- #1

whitejac

- 169

- 0

## 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

**X**i's are i.i.d. with

**EX**

_{i}

**=0**and

**Var(X**

_{i}

**)=4**units. The engineer reports the average of measurements

**M**

_{n}

**= (Y**

_{1}+ ... Y_{n}) / nHow

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(y

_{1}≤ Y ≤ y

_{2})

= P( (y

_{1}- nμ) / (σ√n) ≤ ((X

_{1}+... X

_{n}) - nμ) / (σ√n)) ≤ (y

_{2}- 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.