9C Coreq

In the next preview assignment and in the next class, you will need to find probabilities  and percentiles of a normal distribution.

Empirical Rule

When a variable has a normal distribution, the Empirical Rule will apply:

  • About 68% of the values will fall within one standard deviation from the mean.
  • About 95% of the values will fall within two standard deviations from the mean.
  • About 99.7% of the values will fall within three standard deviations from the mean.

The Scholastic Aptitude Test (SAT) is an assessment designed to evaluate a student’s  college-specific skills. SAT scores tend to follow an approximate normal distribution with a mean of 1060 and a standard deviation of 195.[1]

Question 1

Use the Empirical Rule to complete the following sentences.

  1. About 68% of SAT scores will fall between _____ and _____.
  2. About 95% of SAT scores will fall between _____ and _____.
  3. About 99.7% of SAT scores will fall between _____ and _____.

Question 2

Use your answers from Question 1 to draw a graph of the distribution of SAT scores  with the middle 68%, 95%, and 99.7% marked. Clearly label your x-axis.

Normal Percentiles

The American College Test (ACT) is another assessment designed to evaluate a  student’s college-specific skills. ACT scores tend to follow an approximate normal  distribution with a mean of 20.8 and a standard deviation of 5.8.[2]

A percentile of a distribution is the value at which a certain percentage falls below that  value. For example, the 95th percentile of ACT scores would be the score at which 95%  of students score below.

Question 3

Jane scored a 30 on the ACT.

  1. Calculate the associated normal probability to complete the following  sentence:
    Jane scored above _____% of all students taking the ACT.
  2. What percentile is an ACT score of 30?

Question 4

Find the 30th percentile of ACT scores.

Question 5

Find the associated percentiles to complete the following sentence: 75% of students score between _____ and ______ on the ACT.

Question 6

Julie took the ACT and scored 28. Tim took the SAT and scored 1200. Calculate the  z-score for each student to determine whether Tim or Julie scored in a higher  percentile.

 

  1. National Center for Education Statistics. (2019). Table 226.40 SAT mean scores of high school seniors, standard deviations, and percentage of the graduating class taking the SAT, by state: 2017, 2018, and 2019. https://nces.ed.gov/programs/digest/d19/tables/dt19_226.40.asp ↵
  2. National Center for Education Statistics. (2019). Table 226.50 Number and percentage of graduates taking the ACT test; average scores and standard deviations, by sex and race/ethnicity; and percentage of test takers with selected composite scores and planned fields of postsecondary study: selected years, 1995 through 2018. https://nces.ed.gov/programs/digest/d19/tables/dt19_226.50.asp ↵