Respuesta :
Answer:
Bob score was worse than the mean score.
Bob scored within 2 standard deviations of the mean score.
About 5% of students taking the aptitude test did worse than Bob.
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]z = -1.63[/tex]
So the correct options are:
Bob score was worse than the mean score.
He had a negative z-score, which means that his score was below the mean score.
Bob scored within 2 standard deviations of the mean score.
A zscore of -1.63 means that Bob's score was 1.63 standard deviations below the mean score. So yes, he scored within 2 standard deviations of the mean score.
About 5% of students taking the aptitude test did worse than Bob.
[tex]z = -1.63[/tex] has a pvalue of 0.0516. This means that Bob score is close to the 5th percentile, which means that 5% of the students taking the test scores worse than Bob and 95% did beter. So this option is correct.
