Respuesta :
Answer: 23.89%
Step-by-step explanation:
The empirical rule says that the 95% of the data falls in between two standard deviations of the mean.
Given : GMAT scores are normally distributed and the the average score is approximately [tex]\mu=540[/tex].
Also, 95% of his classmates scored between 400 and 680.
Then, by empirical rule , 95% of data falls in between [tex]mu\pm 2\sigma[/tex]
i.e. [tex]540- 2\sigma=400[/tex] (1)
[tex]540+2\sigma=680[/tex] (2)
Subtracting (1) from (2), we get
[tex]4\sigma=680-400=280\\\\\Rightarrow\ \sigma=\dfrac{280}{4}=70[/tex]
Let x be the random variable to represent the scores of every student.
Statistic z-score : [tex]z=\dfrac{x-\mu}{\sigma}[/tex]
For x= 490, we have
[tex]z=\dfrac{490-540}{70}\approx-0.71[/tex]
The p-value = [tex]P(z<-0.71)=0.2388521\approx23.89\%[/tex]
Hence, 23.89% of his classmates who scored lower than he did .
23.89% of his classmates scored lower than he did.
Given
GMAT scores are normally distributed and the average score is approximate.
Also, 95% of his classmates scored between 400 and 680.
What is the empirical rule?
The empirical rule says that, in a normal data set, virtually every piece of data will fall within three standard deviations.
By empirical rule, 95% of data fall in between;
[tex]\rm= m\pm2\sigma[/tex]
Therefore,
[tex]\rm 540-2\sigma=400\\\\540+2\sigma=680[/tex]
Subtracting (1) from (2), we get
[tex]4\sigma =280\\\\\sigma = \dfrac{280}{4}\\\\\sigma=70[/tex]
Let x be the random variable to represent the scores of every student.
Then,
The value of the z-score is;
[tex]\rm z-score=\dfrac{x-\mu}{\sigma}\\\\ z-score=\dfrac{490-540}{70}\\\\z-score=\dfrac{-50}{70}\\\\z-score=-0.71[/tex]
The value of p is 23.89.
Hence, 23.89% of his classmates scored lower than he did.
To know more about the Empirical formula click the link given below.
https://brainly.com/question/11588623
