91Ó°ÊÓ

• A bar chart or bar graph is a chart or graph that uses rectangular bars with heights or lengths proportionate to the values they represent to convey categorical data. The bars can be plotted either horizontally or vertically.
• A Gaussian's graph has a distinctive symmetric "bell curve" form. The height of the curve's peak is controlled by the parameter a, the position of the centre of the peak is controlled by the parameter b, and the breadth of the "bell" is controlled by the parameter c (the standard deviation, also known as the Gaussian RMS width).

The lifespan of a hypothetical set of incandescent light bulbs are shown by a bar graph and a Gaussian curve.

The standard deviation is given in z multiples of the difference from the mean value.

$z=\frac{x-\mathrm{μ}}{\mathrm{σ}}≈\frac{x-\stackrel{¯}{x}}{s}$

where is the lifetime of individual bulbs,$\stackrel{¯}{x}$ is average mean and s is standard deviation.

The data which are given in Figure 4.l:

$$\begin{gathered} \stackrel{¯}{x}=845.2 \\ \mathrm{σ}=94.2 \end{gathered}$$

The fraction of bulbs when$x=1005.3h$ is:

$$\begin{gathered} z=\frac{1005.3-845.2}{94.2} \\ =1.7 \end{gathered}$$

The area of a range determines the likelihood of measuring in that range.

Table 4-1 shows the area under each part of the Gaussian curve.

The area from the mean to z = 1.7 is 0.4554 , as shown in Table 4 - 1 , ordinate and area for the normal (Gaussian) error curve.

The area above z = 1.7 is 0.5 - 0.4554 = 0.0446 .

As a result, in Figure 4 - 1 , the proportion of bulbs with a lifetime more than 1005.3h is 0.0446 .

(b)

The given data are:

$$\begin{gathered} \stackrel{¯}{x}=845.2 \\ \mathrm{σ}=94.2 \end{gathered}$$

The fraction of bulbs when x = 798.1h is:

$$\begin{gathered} z=\frac{798.1-845.2}{94.2} \\ =-0.5 \end{gathered}$$

The area between the mean and z = -0.5 is the same as the area between the mean and z = +0.5 .

The area from the mean to t = 0.5 is 0.1915 , as shown in Table 4 - 1 , ordinate and area for the normal (Gaussian) error curve.

The fraction of bulbs when x = 901.7h is:

$$\begin{gathered} z=\frac{901.7-845.2}{94.2} \\ =0.6 \end{gathered}$$

The area from the mean to z = 0.6 is 0.2258 , as shown in Table 4 - 1 , ordinate and area for the normal (Gaussian) error curve.

The region between 798.1 and 901.7 is made up of two parts:

0,1915 + 0.2258 = 0.4173

(c)

Using the Excel NORMDIST function, the proportion of bulbs predicted to last between 800 and 900h is determined.

The table below shows the results.