Respuesta :
Answer:
0.73% of the scores are greater than 2317.
14.46% of the scores are less than 1190.
38.23% of the scores are between 1351 and 1673.
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]
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 = 1530, \sigma = 322[/tex]
Find the percentage of scores greater than 2317.
This is 1 subtracted by the pvalue of Z when X = 2317. So:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{2317 - 1530}{322}[/tex]
[tex]Z = 2.44[/tex]
[tex]Z = 2.44[/tex] has a pvalue of 0.9927.
So 1-0.9927 = 0.0073 = 0.73% of the scores are greater than 2317.
Find the percentage of scores less than 1190.
This is the pvalue of Z when X = 1190. So:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{1190 - 1530}{322}[/tex]
[tex]Z = -1.06[/tex]
[tex]Z = -1.06[/tex] has a pvalue of 0.1446.
So 14.46% of the scores are less than 1190.
Find the percentage of scores between 1351 and 1673.
This is the pvalue of Z when X = 1673 subtracted by the pvalue of Z when X = 1351. So
X = 1673
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{1673- 1530}{322}[/tex]
[tex]Z = 0.44[/tex]
[tex]Z = 0.44[/tex] has a pvalue of 0.67
X = 1351
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{1351- 1530}{322}[/tex]
[tex]Z = -0.56[/tex]
[tex]Z = -0.56[/tex] has a pvalue of 0.2877
So 0.67-0.2877 = 0.3823 = 38.23% of the scores are between 1351 and 1673.
