Respuesta :
Answer:
a) 15.87% of the scores are expected to be greater than 600.
b) 2.28% of the scores are expected to be greater than 700.
c) 30.85% of the scores are expected to be less than 450.
d) 53.28% of the scores are expected to be between 450 and 600.
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. Greater than 600
This is 1 subtracted by the pvalue of Z when X = 600. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{600 - 500}{100}[/tex]
[tex]Z = 1[/tex]
[tex]Z = 1[/tex] has a pvalue of 0.8413.
1 - 0.8413 = 0.1587
15.87% of the scores are expected to be greater than 600.
b. Greater than 700
This is 1 subtracted by the pvalue of Z when X = 700. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{700 - 500}{100}[/tex]
[tex]Z = 2[/tex]
[tex]Z = 2[/tex] has a pvalue of 0.9772
1 - 0.9772 = 0.0228
2.28% of the scores are expected to be greater than 700.
c. Less than 450
Pvalue of Z when X = 450. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{450 - 500}{100}[/tex]
[tex]Z = -0.5[/tex]
[tex]Z = -0.5[/tex] has a pvalue of 0.3085.
30.85% of the scores are expected to be less than 450.
d. Between 450 and 600
pvalue of Z when X = 600 subtracted by the pvalue of Z when X = 450. So
X = 600
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{600 - 500}{100}[/tex]
[tex]Z = 1[/tex]
[tex]Z = 1[/tex] has a pvalue of 0.8413.
X = 450
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{450 - 500}{100}[/tex]
[tex]Z = -0.5[/tex]
[tex]Z = -0.5[/tex] has a pvalue of 0.3085.
0.8413 - 0.3085 = 0.5328
53.28% of the scores are expected to be between 450 and 600.