12Ginversenormal


 * Inverse Cumulative Normal Distribution **

Aim: To answer questions of the form: Find **a** so that Pr(X < a) = 0.9

Historical Note:
 * Before calculators, to find a we would have used the CND table backwards.
 * ie Look in the body of the table to find the closest entry to 0.9 and then read off the z-value
 * From the table: Pr (Z < b) = 0.9 gives b = 1.28 (then convert from Z to X)

The Good News
 * Use of the CND table is no longer in the course. Now we get to use calculators.

Using the CAS calculator (Classpad)

... ... These problems can be solved using the** Inverse Cumulative Normal Distribution **. ... ... {Like all Inverse functions, it allows us to go backwards -- in the same way as sqrt is the inverse of sqr, and Arsine is the inverse of Sine}

... ... You will find the ** InvNormCDf ** function in the ** Distribution / Inverse ** entry of the ** Interactive ** menu.

... ... go to the ** statistics ** package and it is in the ** Distribution ** entry of the ** Calc ** menu.
 * Or: **

Tail

... ... The first option on the menu is " Tail "

Using X ~ N( m, s 2 )


 * InvNormCDf( Left, p, s, m )
 * supplies **a** where Pr(X < **a**) = p


 * InvNormCDf( Right, p, s, m )
 * supplies **a** where Pr(X > **a**) = p


 * InvNormCDf( Center, p, s, m )
 * supplies **a** where Pr(**a** < X < **b**) = p
 * {Where **a** and **b** are symmetrical about m }

**Note:** If using __**Center Tail**__, the function in the **Main** package (Interactive menu) only supplies the left limit (**a**). The invnormalcdf function In the **Statistics** package supplies both limits (**a** and **b)**


 * Note: ** Some types of calculators can only calculate the Left Tail.


 * Example 1 **

... Recall that Z ~ N(0, 1)
 * a) ** Find **a** so that Pr(Z < **a**) = 0.9

... ... **enter:** InvNormCDf Tail = Left, p = 0.9, sd = 1, mn = 0

... ... a = 1.2816

... find **b** so that Pr(X < **b**) = 0.7.
 * b) ** Given X ~ N(10, 9) ie s = 3

... ... **enter:** InvNormCDf Tail = Left, p = 0.7, sd = 3, mn = 10

... ... b = 11.5732

... find **c** so that Pr(X > **c**) = 0.65
 * c) ** Given X ~ N(10, 9) ie s = 3

... ... **enter:** InvNormCDf Tail = Right, p = 0.65, sd = 3, mn = 10

... ... c = 8.8440 Note: If your calculator only finds the Left tail, you will need to enter p = 1 – 0.65 = 0.35

... Find **a** so that Pr(12 – **a** < X < 12 + **a**) = 0.4
 * d) ** Given that X ~ N(12, 16) ie s = 4

... ... **enter:** InvNormCDF, Tail = Center, p = 0,4, sd = 4, mn = 12

... ... Note: This returns 9.9024 which represents the left or lower limit of our desired area. ... ... 12 – a = 9.9024

... ... a = 12 – 9.9024

... ... a = 2.0976

... ... The right or upper limit is therefore 12 + 2.0976 = 14.0976

... ... ... This will give the left limit value of 9.9024 ... ... ... If you want to find the upper or right limit, you will need to enter p = 1 – 0.3 = 0.7
 * Note**: If your calculator only finds the Left tail, you will need to enter p = (1 – 0.4)/2 = 0.3. (see graph)


 * Finding an Unknown Mean or Standard Deviation **

This can be done by finding the appropriate Z value that gives the desired probability Then using the rule to convert from Z back to X

math . \qquad z=\dfrac{x - \mu}{\sigma} \qquad. math

**Example 2**

... ... Given X ~ N( 9, s 2 ) ... ... and Pr(X < 12) = 0.8, ... ... Find s


 * Solution: **
 * {We don't know the standard deviation so we can't use the calculator immediately} **

We do know that Z ~ N(0, 1)

so we can find **a** so that Pr(Z < **a**) = 0.8

... ... **enter:** InvNormCDf Tail = Left, p = 0.8, sd = 1, mn = 0

... ... This returns a value of a = 0.8416

... ... Hence Pr(X < 12) = Pr(Z < 0.8416) = 0.8

... ... So when x = 12 we have that z = 0.8416,

Now we can find s using the equation: math \\ . \qquad z=\dfrac{x - \mu}{\sigma} \\. \\ . \qquad 0.8416 = \dfrac{12 - 9}{\sigma} \qquad .\\. \\ . \qquad \sigma = \dfrac{12-9}{0.8416} \\. \\ . \qquad \sigma = 3.56 math

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