Respuesta :
Answer:
The top 20% of the students will score at least 2.1 points above the mean.
Step-by-step explanation:
When the distribution is normal, we use 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.
The mean of a certain test is 14 and the standard deviation is 2.5.
This means that [tex]\mu = 14, \sigma = 2.5[/tex]
The top 20% of the students will score how many points above the mean
Their score is the 100 - 20 = 80th percentile, which is X when Z has a pvalue of 0.8. So X when Z = 0.84.
Their score is:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]0.84 = \frac{X - 14}{2.5}[/tex]
[tex]X - 14 = 0.84*2.5[/tex]
[tex]X = 16.1[/tex]
16.1 - 14 = 2.1
The top 20% of the students will score at least 2.1 points above the mean.
