The test Score of a Math class with 800 students are distributed
normally with a mean of 75 and a standard deviation of 7.
a) What percentage of a class has a test score between 68 and 82?
b) Approximately how many students have a test score between 61
and 89?
c) What is the probability that a student chosen at random has a
test score between 54 and 75?
d) Approximately how many students have a test score greater
than or equal to 96?

Respuesta :

Answer:

  (a) 68.3%

  (b) 764 students

  (c) 49.87%

  (d) 1 student

Step-by-step explanation:

You want to know various percentages and numbers of students in different test score ranges when 800 students have scores normally distributed with a mean of 75 and a standard deviation of 7.

a) [68, 82]

The range 68 to 82 is one standard deviation either side of the mean: 75±7. The "empirical rule" tells you about 68% of students will have a test score in this range. An appropriate calculator gives the fraction as 68.3%.

b) [61, 89]

The range 61 to 89 is 2 standard deviations either side of the mean: 75±2·7. The "empirical rule" tells you about 95% of the 800 students will have a test score in this range. An appropriate calculator puts the number slightly higher than 760 students: 764 students.

c) [54, 75]

The range 54 to 75 is 3 standard deviations below the mean. The empirical rule tells you about 99.7% of the group falls inside 3 standard deviations. Half this number is below the mean, about 49.85% An appropriate calculator puts the number slightly higher: 49.87%.

d) [96, ∞)

The scores at 96 and above are more than 3 standard deviations above the mean. As in part (c), the number in that tail are given by the empirical rule as about 0.15%, or 1.2 students. An appropriate calculator puts the number in the tail of the distribution at a slightly lower value. Hence, the estimated number of students with a score at least 96 is 1 student.

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Additional comment

The numbers in this question seem to be chosen so you can use the empirical rule to find the answers. That rule will make the answers slightly different: 68%, 760 students, 49.85%, 1 student.

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