Respuesta :
Answer:
[tex]P(494.6<X<590)=P(\frac{494.6-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{590-\mu}{\sigma})=P(\frac{494.6-542.3}{120.54}<Z<\frac{590-542.3}{120.54})=P(-0.396<z<0.396)[/tex]
And we can find this probability with this difference:
[tex]P(-0.396<z<0.396)=P(z<0.396)-P(z<-0.396)[/tex]
And in order to find these probabilities we can use tables for the normal standard distribution, excel or a calculator.
[tex]P(-0.396<z<0.396)=P(z<0.396)-P(z<-0.396)=0.654-0.346=0.308[/tex]
Step-by-step explanation:
Assuming that we have the following info: "The admissiones committees for most masters of business administration (MBA) programs requires the Graduate Management Admission Test (GMAT) as part of the application for new students. It has been shown that the scores on GMAT ar enormally distributed with a mean of 542.3 and a standard deviation of 120.54"
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Solution to the problem
Let X the random variable that represent the scores for GAMT of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(542.3,120.54)[/tex]
Where [tex]\mu=542.3[/tex] and [tex]\sigma=120.54[/tex]
We are interested on this probability
[tex]P(494.6<X<590)[/tex]
And the best way to solve this problem is using the normal standard distribution and the z score given by:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
If we apply this formula to our probability we got this:
[tex]P(494.6<X<590)=P(\frac{494.6-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{590-\mu}{\sigma})=P(\frac{494.6-542.3}{120.54}<Z<\frac{590-542.3}{120.54})=P(-0.396<z<0.396)[/tex]
And we can find this probability with this difference:
[tex]P(-0.396<z<0.396)=P(z<0.396)-P(z<-0.396)[/tex]
And in order to find these probabilities we can use tables for the normal standard distribution, excel or a calculator.
[tex]P(-0.396<z<0.396)=P(z<0.396)-P(z<-0.396)=0.654-0.346=0.308[/tex]
You can use the fact that from 2014 to 2017, it was noted that mean of GMAT score is 556.04 points and standard deviation of 120.45 points.
The probability that the sample mean GMAT score for an SRS of n = 15 students will be between 494.6 and 590 is 0.3088
How to get the z scores?
If we've got a normal distribution, then we can convert it to standard normal distribution and its values will give us the z score.
If we have
[tex]X \sim N(\mu, \sigma)[/tex]
then it can be converted to standard normal distribution as
[tex]Z = \dfrac{X - \mu}{\sigma}, \\\\Z \sim N(0,1)[/tex]
That value of Z will be Z score. It will help us to find the probability of the needed region from the z tables.
Using the above method to get the z score and then the needed probability
Suppose the scores in GMAT (the sample taken) be tracked by the random variable X.
Assuming that X's distribution will follow normal distribution, and using the fact that from 2014 to 2017, it was noted that mean of GMAT score is 556.04 points and standard deviation of 120.45 points, we get:
[tex]X \sim N(556.04, 120.45)[/tex]
Having [tex]\mu = 556.04, \sigma = 120.45[/tex]
The needed probability, thus, is given by
[tex]P(494.6 < X < 590)[/tex]
Converting to standard normal distribution, we get:
[tex]Z = \dfrac{X - 556.04}{120.45}, \\\\Z \sim N(0,1)[/tex]
Thus, the needed probability converts to
[tex]P(494.6 < X < 590)\\\\P(\dfrac{494.6 - 556.04}{120.45} < Z < \dfrac{590 - 556.04}{120.45} ) = P(-0.51 < Z < 0.28194)[/tex]
Thus, we have the needed probability as:
[tex]P(-0.51 < Z < 0.28194) = P(Z< 0.28) - P(Z < -0.51) = 0.6103- 0.3015\\ P(-0.51 < Z < 0.28194) \approx 0.3088[/tex]
(values evaluated from z tables)
Thus,
The probability that the sample mean GMAT score for an SRS of n = 15 students will be between 494.6 and 590 is 0.1694
Learn more about standard normal distribution here:
https://brainly.com/question/14989264
