Respuesta :
Answer:
Test scores of 9.7 and less are significantly low.
Test scores of 31.7 and more are significantly high.
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 = 20.7, \sigma = 5.5[/tex]
Significantly low:
Scores of X when Z = -2 and lower. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-2 = \frac{X - 20.7}{5.5}[/tex]
[tex]X - 20.7 = -2*5.5[/tex]
[tex]X = 9.7[/tex]
Test scores of 9.7 and less are significantly low.
Significantly high:
Scores of X when Z = 2 and higher. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]2 = \frac{X - 20.7}{5.5}[/tex]
[tex]X - 20.7 = 2*5.5[/tex]
[tex]X = 31.7[/tex]
Test scores of 31.7 and more are significantly high.
