Notebook

Shallow Examples¶

Example 1

Let X equal the length of life of a 60-watt light bulb marketed by a certain manufacturer. Assume that the distribution of X is $N(\mu, 1296)$. If a random sample of n = 27 bulbs is tested until they burn out, yielding a sample mean of x = 1478 hours, find 95% confidence interval for $\mu$.

Solution:

Here, its given that the population is Normal and also its population SD $\sigma$. So we could use equation \ref{eq:n4} right away.

Given
$\sigma^2 = 1296 \therefore \sigma = 36$,
$x = 1478$, $1-\alpha = 0.95$,
$z_{\frac{\alpha}{2}} = z_{0.025} = 1.96$, $n = 27 \geq 5$

In [1]:

%load_ext tikzmagic

In [2]:

preamble = '''
    \pgfmathdeclarefunction{gauss}{3}{%
      \pgfmathparse{1/(#3*sqrt(2*pi))*exp(-((#1-#2)^2)/(2*#3^2))}%
    }
'''

In [3]:

%%tikz -p pgfplots -x $preamble
% had to be this size to have a normal size in latex 
    \begin{axis}[
        no markers, 
        domain=0:6, 
        samples=100,
        ymin=0,
        axis lines*=left, 
        xlabel=$x$,
        ylabel=$f(x)$,
        height=5cm, 
        width=12cm,
        xtick=\empty, 
        ytick=\empty,
        enlargelimits=false, 
        clip=false, 
        axis on top,
        grid = major,
        axis lines = middle
      ]
    
    \def\mean{3}
    \def\sd{1}
    \def\cilow{\mean - 1.96*\sd}
    \def\cihigh{\mean + 1.96*\sd}
    \addplot [draw=none, fill=yellow!25, domain=\cilow:\cihigh] {gauss(x, \mean, \sd)} \closedcycle;
    \addplot [very thick,cyan!50!black] {gauss(x, 3, 1)};

    \pgfmathsetmacro\valueA{gauss(1,\mean,\sd)}
    \draw [gray] (axis cs:\cilow,0) -- (axis cs:\cilow,\valueA)
    (axis cs:\cihigh,0) -- (axis cs:\cihigh,\valueA);
    \draw [yshift=0.3cm, latex-latex](axis cs:\cilow, 0) -- node [above] {Area = $0.95$} (axis cs:\cihigh, 0);   

    \node[below] at (axis cs:\cilow, 0)  {$\mu - 1.96\sigma$}; 
    \node[below] at (axis cs:\mean, 0)  {$\mu$}; 
    \node[below] at (axis cs:\cihigh, 0)  {$\mu + 1.96\sigma$}; 
    
\end{axis}

$$ \begin{aligned} Pr\Big( x - z_{\frac{\alpha}{2}}\frac{\sigma}{\sqrt{n}} \leq \mu \leq x + z_{\frac{\alpha}{2}}\frac{\sigma}{\sqrt{n}}\Big) = 1-\alpha \nonumber \\ Pr\Big( 1478 - 1.96\frac{36}{\sqrt{27}} \leq \mu \leq 1478 + 1.96\frac{36}{\sqrt{27}}\Big) = 0.95 \nonumber \\ Pr\Big( 1478 - 13.58 \leq \mu \leq 1478 + 13.58\Big) = 0.95 \nonumber \\ Pr\Big( 1464.42 \leq \mu \leq 1491.58\Big) = 0.95 \nonumber \\ \end{aligned} $$

Thus the 95% CI intervals are $[1464.42,1491.58]$. This does not mean, $\mu$ is inside this interval 95% of the time. But simply, if we are to take many such samples and their CIs, 95% of those CIs would contain $\mu$. We do not know what those CIs would be because we do not know the real $\mu$.

test figure¶

just testing if matplotlib output is reduced as well in size in latex..

In [5]:

%matplotlib inline
import matplotlib.pyplot as plt

X = [1,2,3,4,5]
Y = [2,4,6,8,2]

plt.plot(X,Y)
plt.show()