FITTING A NORMAL CURVE

\[1.\ \text{To find the mean μ for the following data of size}\ \hspace{16cm}\\ \text{43,45,47,44,41,46,48,47,45,48,45,42,41,49,42,46,44,49,47,}\ \hspace{15cm}\\ \text{49,41,50,45,45,49,42,46,49,47,46,45,45,49,41,45,49,50,43,}\ \hspace{10cm}\\ \text{45,47,40,45,48,43,42,46,43,45,46,42}\ \hspace{10cm}\]

\[2.\ \text{To find the variance}\ σ^2\ \text{and}\ \hspace{20cm}\\ \text{standard deviation σ}\ \hspace{10cm}\]

\[3.\ \text{To fit the normal curve}\ \hspace{20cm}\\ f(x)\ =\ N(\mu, \sigma^2)\ =\ f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x – \mu)^2}{2\sigma^2}},\ -\ \infty\ \lt\ x\ \lt\ \infty\ \hspace{10cm}\]

\[4.\ \text{To calculate the probability}\ p = P(44 \lt X \lt 47)\ \hspace{18cm}\\ \text{using the formula}\ \int_{X_1}^{X_2} f(x)\ dx\ \hspace{15cm}\]

\[5.\ \text{To calculate the probability}\ p_1\ =\ P(44 \lt X \lt 47)\ \hspace{18cm}\\ \text{using the the probability calculator}\ \hspace{15cm}\\ \text{and verify that}\ p_1\ =\ p\ approximately\ \hspace{10cm}\]

\[6.\ \text{To calculate the number of data points in the interval}\ \hspace{17cm}\\ \text{(44,47) using the the formula}\ n\ =\ N_p\ \hspace{15cm}\\ \text{and verify it with given data}\ \hspace{10cm}\]

\[\color{green}{Step\ 1:}\ \text{Open Geogebra classic 5 (by double clicking on the icon)}\ \hspace{18cm}\]

\[\color{green}{Step\ 2:}\ \text{Open spread sheet view}\ \hspace{19cm}\]

\[\color{green}{Step\ 4:}\ \text{type the following data one by one in cells A1 to A50}\ \hspace{15cm}\\ \text{of column A of the spread sheet}\ \hspace{13cm}\\ \text{43,45,47,44,41,46,48,47,45,48,45,42,41,49,42,46,44,49,47,}\ \hspace{10cm}\\ \text{49,41,50,45,45,49,42,46,49,47,46,45,45,49,41,45,49,50,43,}\ \hspace{10cm}\\ \text{45,47,40,45,48,43,42,46,43,45,46,42}\ \hspace{10cm}\]

\[\color{green}{Step\ 5:}\ \text{To Calculate the mean of the data by using the}\ \hspace{15cm}\\ \text{input command}\ \mu\ =\ mean (A1:A50)\ \hspace{14cm}\]

\[\color{green}{Step\ 6:}\ \text{To Calculate the variance of the data by using the}\ \hspace{15cm}\\ \text{input command σ2 = variance (A1:A50)}\ \hspace{14cm}\]

\[\color{green}{Step\ 7:}\ \text{To Calculate the standard deviation of the data by using the}\ \hspace{15cm}\\ \text{input command σ = sqrt(σ2)}\ \hspace{14cm}\]

\[\color{green}{Step\ 8:}\ \text{To fit the normal curve of the given data}\ \hspace{15cm}\\ f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x – \mu)^2}{2\sigma^2}}\ \hspace{14cm}\\ \text{by using the input bar to type}\ \hspace{15cm}\\ \text{f(x) = ((1)/ (σ sqrt(2 pi))) e^(-((1)/(2))(((x – μ)/(σ)))^(2))}\ \hspace{10cm}\]

\[\color{green}{Step\ 9:}\ \text{To place the cursor near x-axis and right click}\ \hspace{15cm}\\ \text{then choose the option “Zoom to fit”}\ \hspace{16cm}\\ \text{do the same for y-axis}\ \hspace{16cm}\]

\[\color{green}{Step\ 10:}\ \text{To draw the line x = μ by using the input bar}\ \hspace{18cm}\\ \text{to type x = μ}\ \hspace{10cm}\]

\[\color{green}{Step\ 11:}\ \hspace{25cm}\]

\[i.\ \hspace{1cm}\ \text{To mark the point of intersection of the line x = μ}\ \hspace{15cm}\\ \text{and the normal curve by using the intersection tool}\ \hspace{12cm}\]

\[ii.\ \hspace{1cm}\ \text{To mark the point of intersection of the line x = μ}\ \hspace{15cm}\\ \text{with the x-axis by using the intersection tool}\ \hspace{12cm}\]

\[\color{green}{Note:}\ \text{The points of intersection are the points A and B}\ \hspace{18cm}\\ \text{are appearing in the graphics view}\ \hspace{10cm}\]

\[\color{green}{Step\ 12:}\ \text{Deselect the line x = μ in the algebraic view}\ \hspace{18cm}\]

\[\color{green}{Step\ 13:}\ \text{Join the points A and B by using the line segment tool}\ \hspace{15cm}\\ \text{by using input command Mean = segment (A,B)}\ \hspace{14cm}\]

\[\color{green}{Step\ 14:}\ \text{Create sliders for}\ X_1\ and\ X_2\ \hspace{15cm}\\ \text{and set the values}\ X_1\ =\ 44\ and\ x_2\ =\ 47\ \hspace{10cm}\]

\[\color{green}{Step\ 15:}\ \text{Calculate the probability}\ p\ =\ \int_{X_1}^{X_2} f(x)\ dx\ \hspace{15cm}\\ \text{by using the input bar to type p = Integral(f, X_1,X_2)}\ \hspace{10cm}\]

\[\color{green}{Step\ 16:}\ \text{Create a slider for N}\ \hspace{18cm}\\ \text{and set the value N=50(the data size)}\ \hspace{10cm}\]

\[\color{green}{Step\ 17:}\ \text{Calculate the theoretical value of number of data}\ \hspace{15cm}\\ \text{between 44 and 47 using the command}\ \hspace{14cm}\\ \text{n=N p and round – off it to the nearest integer value.}\ \hspace{13cm}\]

\[\color{green}{Note:}\ \text{Count the number of data – points}\ x_i\ such\ that\ 44\ \leq\ x_i\ \leq\ 47,\ \hspace{15cm}\\ \text{from the given data and compare it with n}\ \hspace{10cm}\]

\[\color{green}{Step\ 18:}\ \text{Go to view and open Probability calculator}\ \hspace{15cm}\\ \text{Choose the Normal under the distribution}\ \hspace{10cm}\]

\[\color{green}{Step\ 18:}\ \text{Enter the values of μ and σ (which are 45.34 and 2.69,}\ \hspace{15cm}\\ \text{respectively, for the above data}\ \hspace{10cm}\]

\[\color{green}{Step\ 19:}\ \text{Choose closed interval option []. Enter 44 in the left blank}\ \hspace{15cm}\\ \text{and 47 in the right blank and calculate the probability.}\ \hspace{14cm}\\ \text{Verify that this probability is approximately equal to}\ \hspace{13cm}\\ \text{the probability calculated in the step 15}\ \hspace{11cm}\]