Respuesta :
Answer:
a) 79.10% probability that a student had a score less than 320.
b) 46.63% probability that a student had a score between 250 and 300.
c) 99.25% of the students had a test score greater than 200
d) 350.865
e) 265.025
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 = 290, \sigma = 37[/tex]
a) Find the probability that a student had a score less than 320.
This is the pvalue of Z when X = 320. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{320 - 290}{37}[/tex]
[tex]Z = 0.81[/tex]
[tex]Z = 0.81[/tex] has a pvalue of 0.7910
79.10% probability that a student had a score less than 320.
b) Find the probability that a student had a score between 250 and 300.
This is the pvalue of Z when X = 300 subtracted by the pvalue of Z when X = 250.
X = 300
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{300 - 290}{37}[/tex]
[tex]Z = 0.27[/tex]
[tex]Z = 0.27[/tex] has a pvalue of 0.6064
X = 250
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{250 - 290}{37}[/tex]
[tex]Z = -1.08[/tex]
[tex]Z = -1.08[/tex] has a pvalue of 0.1401
0.6064 - 0.1401 = 0.4663
46.63% probability that a student had a score between 250 and 300.
c) What percent of the students had a test score greater than 200?
This is 1 subtracted by the pvalue of Z when X = 200. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{200 - 290}{37}[/tex]
[tex]Z = -2.43[/tex]
[tex]Z = -2.43[/tex] has a pvalue of 0.0075
1 - 0.0075 = 0.9925
99.25% of the students had a test score greater than 200
d) What is the lowest score that would still place a student in the top 5% of the scores?
X when Z has a pvalue of 1-0.05 = 0.95. So X when Z = 1.645.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.645 = \frac{X - 290}{37}[/tex]
[tex]X - 290 = 37*1.645[/tex]
[tex]X = 350.865[/tex]
e) What is the highest score that would still place a student in the bottom 25% of the scores
X when Z has a pvalue of 0.25. So X when Z = -0.675
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-0.675 = \frac{X - 290}{37}[/tex]
[tex]X - 290 = 37*(-0.675)[/tex]
[tex]X = 265.025[/tex]
