Respuesta :
Answer:
a) 25.14% percentage of the scores expected to be greater than 600.
b) 7.49% percentage of the scores expected to be greater than 700.
c) 31.56% of the scores are expected to be less than 450.
d) 43.3% of the scores are expected to be between 450 and 600.
Step-by-step explanation:
Problems of normally distributed samples can be 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]
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. Subtracting 1 by the pvalue, we This p-value is the probability that the value of the measure is greater than X.
For this problem, we have that:
The College Boards, which are administered each year to many thousands of high school students, are scored so as to yield a mean of 513 and a standard deviation of 130, so [tex]\mu = 513, \sigma = 130[/tex].
What percentage of the scores can be expected to satisfy each of the following conditions?
a) Greater than 600
This is 1 subtracted by the pvalue of Z when [tex]X = 600[/tex]
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{600 - 513}{130}[/tex]
[tex]Z = 0.67[/tex]
[tex]Z = 0.67[/tex] has a pvalue of 0.7486.
So there is a 1 - 0.7486 = 0.2514 = 25.14% percentage of the scores expected to be greater than 600.
b) Greater than 700
This is 1 subtracted by the pvalue of Z when [tex]X = 700[/tex]
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{700 - 513}{130}[/tex]
[tex]Z = 1.44[/tex]
[tex]Z = 1.44[/tex] has a pvalue of 0.9251.
So there is a 1 - 0.9251 = 0.0749 = 7.49% percentage of the scores expected to be greater than 700.
c. Less than 450
This is the pvalue of Z when [tex]X = 450[/tex]
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{450 - 513}{130}[/tex]
[tex]Z = -0.48[/tex]
[tex]Z = -0.48[/tex] has a pvalue 0.31561.
So, 31.56% of the scores are expected to be less than 450.
d. Between 450 and 600
This is the subtraction of the pvalue of [tex]X = 600[/tex] by the pvalue of [tex]X = 450[/tex]
So
[tex]P = 0.7486 - 0.31561 = 0.433[/tex]
43.3% of the scores are expected to be between 450 and 600.
