# -7.6.69 Determine the value z^* that...

• MHB
• karush
In summary: This means that 1.88 is the z-score that corresponds to a probability of 0.97 under the standard normal distribution.
karush
Gold Member
MHB
Determine the value $z^*$ that...
a. Separates the largest $3\%$ of all z values from the others
$=1.88$
b. Separates the largest $1\%$ of all z values from the others
$=2.33$
c. Separates the smallest $4\%$ of all z values from the others
$=1.75$
d. Separates the smallest $10\%$ of all z values from the others
$=1.28$

OK just can't seem to find an example of how these are stepped thru
the book answer follows the =

once again $$\displaystyle \Phi(z)$$ is the CDF of the standard normal distribution
you'll need some way to compute the inverse of this function to complete this problem.

a)$$\displaystyle \Phi(z^*) = 0.97\\ z^* = \Phi^{-1}(0.97)= 1.88079$$

b) $$\displaystyle \Phi(z^*) = 0.99$$

c) $$\displaystyle \Phi(z^*) = 0.04$$

d)$$\displaystyle \Phi(z^*) = 0.1$$

romsek said:
once again $$\displaystyle \Phi(z)$$ is the CDF of the standard normal distribution
you'll need some way to compute the inverse of this function to complete this problem.

a)$$\displaystyle \Phi(z^*) = 0.97\\ z^* = \Phi^{-1}(0.97)= 1.88079$$

b) $$\displaystyle \Phi(z^*) = 0.99$$

c) $$\displaystyle \Phi(z^*) = 0.04$$

d)$$\displaystyle \Phi(z^*) = 0.1$$

mahalo I was unaware of the use of that symbol

ok I can see that at 1.88 goes to .97 on table
or using P to z calculator but still what is $\Phi^{-1}$

Last edited:
$$\displaystyle \Phi^{-1}(p)$$ is the inverse of $$\displaystyle \Phi(z)$$

If you are given a probability $$\displaystyle p, \Phi^{-1}(p)$$ returns the associated z-score of $$\displaystyle p$$

Since $$\displaystyle \Phi(1.88)= 0.97$$, $$\displaystyle \Phi^{-1}(0.97)= 1.88$$

## 1. What does the value z^* represent in the equation -7.6.69 Determine the value z^* that...?

The value z^* represents the critical value in a statistical test, which is used to determine the significance of a result. It is typically compared to the test statistic to determine if the result is statistically significant.

## 2. How is the value z^* calculated?

The value z^* is calculated using a standard normal distribution table or a statistical software program. It is based on the desired level of significance, the sample size, and the type of statistical test being performed.

## 3. What is the significance of determining the value z^*?

Determining the value z^* allows researchers to determine the probability of obtaining a result by chance. This helps to determine the validity and reliability of the results and whether they can be generalized to a larger population.

## 4. Can the value z^* change in different statistical tests?

Yes, the value z^* can change depending on the type of statistical test being performed and the desired level of significance. It is important to use the correct value z^* for the specific test being conducted.

## 5. How does the value z^* relate to the p-value?

The value z^* is used to calculate the p-value, which represents the probability of obtaining a result as extreme or more extreme than the observed result, assuming the null hypothesis is true. The p-value is then compared to the desired level of significance to determine the significance of the result.

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