Answer :
Answer:
Applicants need a score of at least 71.72.
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 = 58, \sigma = 7[/tex]
Top 2.5%.
Scores that are X when Z has a pvalue of 1-0.025 = 0.975 and higher. So it is scores that are X when Z = 1.96 and higher.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.96 = \frac{X - 58}{7}[/tex]
[tex]X - 58 = 7*1.96[/tex]
[tex]X = 71.72[/tex]
Applicants need a score of at least 71.72.
Applicants need a score of at least 71.72.
Given information:
The exam is constructed such that the scores are normally distributed with an average of 58 and a standard deviation of 7.
Calculation of the score:
[tex]Z = x - \mu \div \sigma\\\\1.96 = x - 58 \div 7\\\\x - 58 = 7 \times 1.96\\\\[/tex]
x = 71.72
learn more about the standard deviation here: https://brainly.com/question/2748723