Respuesta :
Answer:
a) 604
b) 628
c) 705.5
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]
a. What SAT score (i.e., X score) separates the top 15% of the distribution from the rest?
This is the value of X when Z has a pvalue of 0.85. So it is X when Z = 1.04.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.04 = \frac{X - 500}{100}[/tex]
[tex]X - 500 = 100*1.04[/tex]
[tex]X = 604[/tex]
b. What SAT score (i.e., X score) separates the top 10% of the distribution from the rest?
This is the value of X when Z has a pvalue of 0.90. So it is X when Z = 1.28.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.28 = \frac{X - 500}{100}[/tex]
[tex]X - 500 = 100*1.28[/tex]
[tex]X = 628[/tex]
c. What SAT score (i.e., X score) separates the top 2% of the distribution from the rest?
This is the value of X when Z has a pvalue of 0.98.So it is X when Z = 2.055.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]2.055 = \frac{X - 500}{100}[/tex]
[tex]X - 500 = 100*2.055[/tex]
[tex]X = 705.5[/tex]
- A) the SAT score that separates the top 15% of the distribution from the rest is 510;
- B) the SAT score that separates the top 10% of the distribution from the rest is 540; and
- C) the SAT score that separates the top 2% of the distribution from the rest is 588.
Standard deviation
Given that the distribution of SAT scores is normal with a mean of µ = 500 and a standard deviation of σ = 100, to determine A) what SAT score separates the top 15% of the distribution from the rest; B) what SAT score separates the top 10% of the distribution from the rest; and C) what SAT score separates the top 2% of the distribution from the rest, the following calculations must be made:
A)
- (500 + 100) / 100 x (100 - 15) = X
- 600 / 100 x 85 = x
- 6 x 85 = X
- 510 = X
B)
- (500 + 100) / 100 x (100 - 10) = X
- 600 / 100 x 10 = X
- 6 x 90 = x
- 540 = X
C)
- (500 + 100) / 100 x (100 - 2) = X
- 600 / 100 x 98 = X
- 6 x 98 = x
- 588 = X
Therefore, A) the SAT score that separates the top 15% of the distribution from the rest is 510; B) the SAT score that separates the top 10% of the distribution from the rest is 540; and C) the SAT score that separates the top 2% of the distribution from the rest is 588.
Learn more about standard deviation in https://brainly.com/question/12402189
