-7.68 Determine the value z* to satisfy the following conditions

In summary, z score is a number that is used to compare two normal distributions. The z score is calculated by multiplying the probability of the difference between the two distributions and the standard deviation of the difference between the two distributions. The z score is then divided by the probability of the difference between the two distributions.
  • #1
karush
Gold Member
MHB
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5
$\tiny{7.68}$
Normal curve $\quad\mu=12\quad\sigma = 2$
Let z denote a variable that a standard normal distribution.
Determintie the value $z*$ to satisfy the following conditions:
a. $P(z<z*)=.025$
b. $P(z<z*)=.01$
c. $P(z<z*)=.05$
d. $P(z<z*)=.02$
e. $P(z>z*)=.01$
f. $P(z>z*)\textbf{ or }(z<-z*)=.20$

ok I want to do these one at a time especially a,d and f

example on page 411 of text
SSCtw.png
 
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  • #2
\(\displaystyle \text{let $\Phi(z)$ be the CDF of the standard normal. That function they make tables of. You'll need some form of that.}\\~\\

\text{Then $\Phi\left(\dfrac{z - \mu}{\sigma}\right)$ is the CDF of a normal distribution centered at $\mu$ with standard deviation $\sigma$}\\~\\

\text{So $P(z < z^*) = \Phi\left(\dfrac{z^*-\mu}{\sigma}\right)$}\)

Similarly

\(\displaystyle P(z > z^*) = 1-\Phi\left(\dfrac{z^*-\mu}{\sigma}\right)\)

Finally for (f) the probability of two non-intersecting intervals is the sum of the individual interval probabilities.

You'll actually have to use the inverse function \(\displaystyle \Phi^{-1}(p)\) as they give you probabilities.

For example (a)

\(\displaystyle P(z < z^*) = 0.25\\
\text{we look up in the table $0.25$ (actually I just use software) and find that it corresponds to a "z-score", as it's commonly known, of $-0.67449$}\\
\dfrac{z^* - \mu}{\sigma} = -0.67449\\~\\
z^* = \mu - 0.67449\sigma = 10.651\)
 
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  • #3
mahalo
ill try the rest
romsek said:
\(\displaystyle \text{let $\Phi(z)$ be the CDF of the standard normal. That function they make tables of. You'll need some form of that.}\\
\text{Then $\Phi\left(\dfrac{z - \mu}{\sigma}\right)$ is the CDF of a normal distribution centered at $\mu$ with standard deviation $\sigma$}\\
\text{So $P(z < z^*) = \Phi\left(\dfrac{z^*-\mu}{\sigma}\right)$}\)
Similarly
\(\displaystyle P(z > z^*) = 1-\Phi\left(\dfrac{z^*-\mu}{\sigma}\right)\)
Finally for (f) the probability of two non-intersecting intervals is the sum of the individual interval probabilities.
You'll actually have to use the inverse function \(\displaystyle \Phi^{-1}(p)\) as they give you probabilities.
For example (a)
\(\displaystyle P(z < z^*) = 0.25\\
\text{we look up in the table $0.25$ (actually I just use software) and find that it corresponds to a "z-score", as it's commonly known, of $-0.67449$}\\
\dfrac{z^* - \mu}{\sigma} = -0.67449\\
z^* = \mu - 0.67449\sigma = 10.651\)

mahalo
ill try the rest
 
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  • #4
Determintie the value z∗z∗ to satisfy the following conditions

[c] $P(z<z*)=.05$
$\quad\dfrac{z^* - \mu}{\sigma} =-5.975 \textbf{ so } z^*=(12)-5.975(2)=.05$

did something wrong...

z score calc
 
  • #5
use calculate z from p
 

Related to -7.68 Determine the value z* to satisfy the following conditions

1. What is the value of z*?

The value of z* is a numerical variable that satisfies the given conditions.

2. How do you determine the value of z*?

The value of z* can be determined by solving the given conditions using mathematical equations and techniques.

3. What are the conditions that need to be satisfied to determine the value of z*?

The conditions that need to be satisfied include the given numerical value of -7.68 and any additional information provided in the problem.

4. Can the value of z* be negative?

Yes, the value of z* can be negative depending on the given conditions and the mathematical solution to satisfy those conditions.

5. Is there a specific formula to calculate the value of z*?

There is no specific formula to calculate the value of z*. It depends on the given conditions and requires the use of mathematical equations and techniques to solve for the value of z*.

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