# Normal distribution graph P(a<x<b)

• MHB
• karush
In summary, the probability that the lifespan of an insect of this species lies between 55 and 60 hours is represented by the shaded area in the following diagram.
karush
Gold Member
MHB
The probability that the lifespan of an insect of this species lies between 55 and 60 hours is represented by the shaded area in the following diagram.\\
Write down the values of a and b.
$a=\dfrac{2}{4.4}= 0.455 b=\dfrac{3}{4.4}=0.682]$

ok this was a key to a test question from 2013 but mostly I wanted to use tikz to plot the graph
also not real sure why they used the calculation they did 2 and 3 ?

no sure why this did not render here it was fine in Overleaf
also not sure even if the graph is correct
also the word noramcdf showed up on the PDF is that a mathlab or something

%preamble \usepackage{pgfplots}
\newcommand\gauss[2]{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))} % Gauss function, parameters mu and sigma
\begin{axis}[every axis plot post/.append style={
mark=none,samples=50,smooth}, % All plots: 50 samples, smooth, no marks
axis x line*=bottom, % no box around the plot, only x axis
axis y line=none, % the * suppresses the arrow tips
enlargelimits=upper, % extend the axes a bit to the right and top
domain=-2:3, % Default for all plots: from -4:4
xtick={-.455,.655},
xticklabels={$-.455$,$.655$},
width=10cm,
height=4cm]
\addplot [fill=gray!30, draw=none, domain=-0.45:0.655] {\gauss{-0.455}{0.655}} \closedcycle;
\end{axis}
\end{tikzpicture}

Last edited:
It's because \begin{tikzpicture} is missing at the beginning.
When we add it, we get:
\begin{tikzpicture}
%preamble \usepackage{pgfplots}
\newcommand\gauss[2]{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))} % Gauss function, parameters mu and sigma
\begin{axis}[every axis plot post/.append style={
mark=none,samples=50,smooth}, % All plots: 50 samples, smooth, no marks
axis x line*=bottom, % no box around the plot, only x axis
axis y line=none, % the * suppresses the arrow tips
enlargelimits=upper, % extend the axes a bit to the right and top
domain=-2:3, % Default for all plots: from -4:4
xtick={-.455,.655},
xticklabels={$-.455$,$.655$},
width=10cm,
height=4cm]
\addplot [fill=gray!30, draw=none, domain=-0.45:0.655] {\gauss{-0.455}{0.655}} \closedcycle;
\end{axis}
\end{tikzpicture}

I should of seen that,,,,ok I added to the OP

## 1. What is a normal distribution graph?

A normal distribution graph is a bell-shaped curve that represents a probability distribution of a continuous random variable. It is symmetric around the mean and has a specific area under the curve that corresponds to the probability of a random variable falling within a certain range.

## 2. How is a normal distribution graph calculated?

A normal distribution graph is calculated using a mathematical formula that takes into account the mean (μ) and standard deviation (σ) of the data. The formula is f(x) = (1/σ√2π) * e^(-(x-μ)^2/2σ^2), where e is the base of the natural logarithm.

## 3. What does P(a<x<b) mean in a normal distribution graph?

P(a<x<b) represents the probability of a random variable falling between a and b on the normal distribution graph. This is also known as the cumulative probability and can be calculated by finding the area under the curve between the two points.

## 4. What does the area under a normal distribution graph represent?

The area under a normal distribution graph represents the probability of a random variable falling within a certain range. For example, the area under the curve between the mean and one standard deviation above the mean represents the probability of the random variable falling within that range.

## 5. What is the significance of the 68-95-99.7 rule in a normal distribution graph?

The 68-95-99.7 rule, also known as the empirical rule, states that approximately 68% of the data falls within one standard deviation of the mean, 95% of the data falls within two standard deviations of the mean, and 99.7% of the data falls within three standard deviations of the mean in a normal distribution graph. This rule helps to understand the spread of data and identify outliers.

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