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Algebra II & Trigonometry                                          Normal Distribution

A2.S.5          Know and apply the characteristics of the normal distribution

In 1 – 4, in each case, write the number that makes the sentence true.
1. In a normal distribution, the mean is at the __________ percentile.

2. Approximately __________% of the values in a normal distribution lie within one standard deviation of the mean.

3. A value that is one standard deviation below the mean in a normal distribution is at the __________ percentile.

4. In a normal distribution, about __________% of the values lie in the interval from x  2 to x  2 .

5. For the scores on a standardized test, the mean is 82 and the standard deviation is 6. Find the scores for
a. x  2                       b. x                            c. x                            d. x  2

In 6 – 12, select the numeral preceding the choice that best completes the statement or answers the question.
6. On a standardized test, the mean score is 70 and the standard deviation is 10. If Rita Keane scored 90 on this test, her
score is at the
(1) 84th percentile               (2) 90th percentile             (3) 95th percentile              (4) 97.5th percentile

7. If the mean score of an IQ test is 104 and the standard deviation is 7, then 68% of the scores are
(1) less than 104                (2) between 90 and 104            (3) between 97 and 111            (4) greater than 104

8. Mr. Noren bowls regularly, and his average score is 182. His bowling scores approximate a normal distribution with a
standard deviation of 8.5. He can expect to bowl between 182 and 199
(1) about 47.5% of the time.                             (2) about 34% of the time.
(3) more than 50% of the time.                           (4) less than 16% of the time.

9. The lengths of time for telephone calls approximate a normal distribution. If the mean length is 4.3 minutes with a
standard deviation of 1.4 minutes, about 84% of the calls are
(1) 4.3 minutes or less                                   (2) 5.7 minutes or less
(3) from 2.9 to 5.7 minutes                               (4) 4.3 minutes or more

10. For a certain group of students, the mean score on a college aptitude test was 430 with a standard deviation of 108.
(a) Which of the following interval of scores should occur most frequently?
(1) below 322            (2) between 214 and 332           (3) between 322 and 538         (4) above 430
(b) Which of the following scores might be at the 70th percentile?
(1) 505                  (2) 540                           (3) 590                         (4) 700

11. Ms. Surber owns a real-estate business. Sale prices of homes in her area approximate a normal distribution, with a
mean of \$72,000 and a standard deviation of \$7,300. A home that sells for \$87,600 would rank:
(1) below the 75th percentile                           (2) between the 75th percentile and the 85th percentile
(3) between the 85  th percentile and 95th percentile   (4) above the 94th percentile

12. In a contest, the mean time for pianists to play the “Minute Waltz” was 56 seconds, with a standard deviation of 3
seconds. The times approximated a normal distribution. If Al Cavallaro played the “Minute Waltz” in 50 seconds, his time
was faster than what percent of the population?
(1) 2.5%                         (2) 16%                           (3) 84%                          (4) 97.5%
13. In a large school district, the years of service for the teaching staff approximate a normal distribution. The mean is 16
years of teaching, and the standard deviation is 7.4 years. Four teachers, and the number of years of service for each, are
listed below.
Robert Novak: 16 years                        Mabel Oestrich: 35 years
Samuel Backer: 21 years                       Nancy Garbowski: 12 years
According to years of service, name the teachers(s), if any, who might possibly rank at the
a. 10th percentile                          b. 35th percentile                         c. 50th percentile

d. 70th percentile                        e. 90th percentile                        f. 99th percentile

14. In the diagram, about 95% of the data values fall in the shaded area.
If the distribution is normal and these values range from 25 to 51, find:
a. the standard deviation
b. the mean

In 15 – 18, in each case assume a normal distribution. Using the given
information, find:
a. the standard deviation              b. the mean
15. x    75 and x    109

16. x    4.5 and x  2  25.5

17. Values within one standard deviation of the mean range from 9.2 to 14.

18. About 2.5% of the values are at or below 45, and 2.5% are above 77.

19. In a normal distribution, x  27.9 , and 16% of the values are at or below 25.2. Find:
a.                               b. x  2

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