View source for Resource assessment exercises: required sample size determination

: ''This article is part of the '''Resource assessment exercises'''. See the [[:category:Resource assessment exercises 2014|category page]] for a (chronological) table of contents.

[[Resource assessment exercises: standard error and confidence intervals|Previously]] we took a sample <code>S</code> of size <math>n=50</math>.

<pre>
S

##  [1] 22  8 18 43 21 44 17 25 32 10 11 17  9 10 56 14 14 10 20  8 37 14 55 29 33
## [26] 17 10 15 29  8 21  9  9 24 21 28 19 58 16 16 15 20  5  9 14 30 11  9 12 27
</pre>

The width of the confidence interval was,

<pre>
sd(S)/sqrt(n) * qt(0.975, n - 1) * 2

## [1] 7.364
</pre>

Suppose a confidence interval of <math>A=3</math> cm is desired. How large should the sample size, <math>n</math>, be? This can be estimated,

{{EquationRef|equation=$A=t_{\alpha,n-1}\frac{s}{\sqrt{n} }\rightarrow n=\frac{t_{\alpha, n-1}^2s^2}{A^2}$|1}}

We will use the sample <code>S</code> (<math>n=50</math>) to estimate how many observations we need in our sample. 
In [[:wikipedia:R_(programming_language)|R]]:

<pre>
half.A <- 1.5
req.n <- (qt(0.975, n - 1)^2 * var(S))/half.A^2
req.n

## [1] 301.2
</pre>

We always need to round up!

<pre>
req.n <- ceiling(req.n)
</pre>

{{info|message=What the function <code>ceiling()</code> does|text=The function <code>ceiling()</code> ceiling takes a single numeric argument <code>x</code> and returns a numeric vector containing the smallest integers not less than the corresponding elements of <code>x</code> E.g., 4.3 becomes 5. See also <code>round()</code>.}}
  

We estimate the width of the confidence interval using the new sample size of <math>n=302</math>.

<pre>
s.req.n <- sample(trees$dbh, req.n)
sd(s.req.n)/sqrt(req.n) * qt(0.975, req.n - 1) * 2

## [1] 2.759
</pre>

==Related articles==
* Previous article: [[Resource assessment exercises: finite population correction|Finite population correction]]
* Next article: [[Resource assessment exercises: estimating a proportion|Estimating a proportion]]

[[category:Resource assessment basics in R (2014)|Required sample size determination]]