Respuesta :
Answer:
The minimum score needed to be accepted is 525.3.
Step-by-step explanation:
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this problem, we have that:
[tex]\mu = 500, \sigma = 100[/tex]
If the state college only accepts students who score in the top 60% on the SAT, what is the minimum score needed to be accepted?
The 60th percentile, which is the value of X when Z has a pvalue of 0.6. So it is X when Z = 0.253. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]0.253 = \frac{X - 500}{100}[/tex]
[tex]X - 500 = 0.253*100[/tex]
[tex]X = 525.3[/tex]
The minimum score needed to be accepted is 525.3.
The minimum score needed to be accepted is 525.3.
The first thing to solve a normal distribution problem is to find out the z-score.
What is a z-score?
A Z-score is a numerical measurement that describes a value's relationship to the mean of a group of values.
Formula to find out z-score
[tex]Z=\frac{X-A}{B}[/tex]
where,
X = observed value
A = mean value of the sample
B = standard deviation of the sample.
After we find the Z-score, we need to watch the z-score table and find the p-value associated with this z-score.
The p-value is the probability that the measured value is less than X.
It is given that
State college only accepts students who score in the top 60% on the SAT means p=0.60
From the table, the z-score that corresponds to the p-value 0.60 is 0.253
Now by using the above formula,
[tex]0.253=\frac{X-500}{100}\\\\X= 500+25.3=525.3[/tex]
The minimum score needed to be accepted is 525.3.
To get more about normal distribution problems refer to the link,
https://brainly.com/question/4079902
