91Ó°ÊÓ

It has been given that 4 letters are put at random into 4 envelopes

Probability means the chances of any event to occur is called it probability.

Let$X~N(\mathrm{μ},{\mathrm{σ}}^2).$The precent of the confidence interval$\mathrm{μ}Â\pm h$is equal to the expression mentioned below.

Here$Z~N(0,1)$and$\mathrm{Φ}$is the distribution function of Z. Use the programming language (like R) we can calculate the required. For example, the limits$90\%$confidence interval are equal to the expression derived below.

$\begin{array}{c}2\mathrm{Φ}\left(\frac{h}{\mathrm{σ}}\right)−1=0.9\\ \mathrm{Φ}\left(\frac{h}{\mathrm{σ}}\right)=0.95\\ h=1.6449\mathrm{σ}\end{array}$

Here the code is 0.95 similarly, the limits for$95\%$confidence interval is mentioned below.

data-custom-editor="chemistry" $\begin{array}{c}2\mathrm{Φ}\left(\frac{h}{\mathrm{σ}}\right)−1=0.95\\ \mathrm{Φ}\left(\frac{h}{\mathrm{σ}}\right)=0.975\\ h=1.96\mathrm{σ}\end{array}$

The limits$for99\%$confidence interval is given below.

$\begin{array}{c}2\mathrm{Φ}\left(\frac{h}{\mathrm{σ}}\right)−1=0.99\\ \mathrm{Φ}\left(\frac{h}{\mathrm{σ}}\right)=0.995\\ h=2.5758\mathrm{σ}.\end{array}$

Plug $h=1.3\mathrm{σ}$, the confidence of that interval is equal to the value given below.

$\begin{array}{c}2\mathrm{Φ}\left(\frac{1.3\mathrm{σ}}{\mathrm{σ}}\right)−1=2\mathrm{Φ}\left(1.3\right)−1\\ =0.8064\end{array}$