Respuesta :
Exercise 1
We have the following data for the applicants:
• their scores are normally distributed,
,• the mean score is μ = 63,
,• the standard deviation is σ = 7,
,• the number of applicants is n = 1000.
1) How many applicants likely scored higher than 70 on the ARE?
The number of students that will score higher than 70 is given by:
[tex]N=n\cdot P(x\ge70),[/tex]where:
• n is the total number of students,
,• P(x ≥ 70) is the probability that a student gets a score x ≥ 70.
To compute the probability needed, we compute the corresponding z-score for this case:
[tex]z=\frac{x-\mu}{\sigma}=\frac{70-63}{7}=1._{}[/tex]Using the properties of the normal distribution, and table for the z-scores, we have that:
[tex]P(x\ge70)=P(z\ge1)=0.1587.[/tex]Replacing in the formula above, the number of students that will score higher than 70 is:
[tex]N=n\cdot P(x\ge70)=1000\cdot0.1587=158.7\cong159.[/tex]2) 95% of applicants could be expected to score within what range?
To answer this question, we use the Empirical Rule for normal distributions: "The Empirical Rule states that 99.7% of data observed following a normal distribution lies within 3 standard deviations of the mean. Under this rule, 68% of the data falls within one standard deviation, 95% within two standard deviations, and 99.7% within three standard deviations from the mean".
In this case, the rule states that we expect to have 95% of applicants with scores within the following range of scores:
[tex](\mu-2\sigma,\mu+2\sigma)=(63-2\cdot7,63+2\cdot7)=(49,77)\text{.}[/tex]3) Any applicants who score higher than 84 on the ARE are awarded a certificate of achievement. For the upcoming school year, how many such certificates will be needed?
The number of students that will score higher than 84 is given by:
[tex]N=n\cdot P(x\ge84)\text{.}[/tex]We compute the probability as before. The z-score for this case is:
[tex]z=\frac{x-\mu}{\sigma}=\frac{84-63}{7}=3.[/tex]Using a table for z-scores, we find that the probability is:
[tex]P(x\ge84)=P(z\ge3)=0.0013.[/tex]Replacing in the formula above, we find that the number of certificates needed for students with scores higher than 84 is:
[tex]N=n\cdot P(x\ge84)=1000\cdot0.0013=1.3\cong1.[/tex]Answers
0. The number of applicants likely to score higher than 70 on the exam is 159.
,1. It is expected to have 95% of applicants with scores in the range: 49 ≤ x ≤ 77.
,2. It will be needed 1 certificate for the upcoming school year.
