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

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In summary: This means that 1.88 is the z-score that corresponds to a probability of 0.97 under the standard normal distribution.
  • #1
karush
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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 =
 
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  • #2
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)\(\displaystyle \Phi(z^*) = 0.1\)
 
  • #3
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

Screenshot 2021-09-02 11.25.39 AM.png


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:
  • #5
\(\displaystyle \Phi^{-1}(p)\) is the inverse of \(\displaystyle \Phi(z)\)

If you are given a probability \(\displaystyle p, \Phi^{-1}(p)\) returns the associated z-score of \(\displaystyle p\)
 
  • #6
Since \(\displaystyle \Phi(1.88)= 0.97\), \(\displaystyle \Phi^{-1}(0.97)= 1.88\)
 

Related to -7.6.69 Determine the value z^* that...

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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