Mrs. Smith's Algebra 2 class scored very well on the last quiz. With one exception, everyone received an A. Within how many standard deviations of the mean do all the quiz grades fall? 97, 95, 97, 94, 93, 99, 91, 96, 94, 88, 92, 91 (1 point) Responses 3 3 4 4 2 2 1
To find how many standard deviations away from the mean all the quiz grades fall, we first need to find the mean and standard deviation of the quiz grades.
Mean:
(97 + 95 + 97 + 94 + 93 + 99 + 91 + 96 + 94 + 88 + 92 + 91) / 12 = 1050 / 12 = 87.5
Standard deviation:
Calculate the variance first by finding the squared differences between each grade and the mean, then average them:
(97 - 87.5)^2 + (95 - 87.5)^2 + (97 - 87.5)^2 + (94 - 87.5)^2 + (93 - 87.5)^2 + (99 - 87.5)^2 + (91 - 87.5)^2 + (96 - 87.5)^2 + (94 - 87.5)^2 + (88 - 87.5)^2 + (92 - 87.5)^2 + (91 - 87.5)^2 = 452.5
Variance = 452.5 / 12 = 37.7083
Standard deviation = square root of the variance = √37.7083 ≈ 6.14
All quiz grades fall within:
87.5 ± 2 * 6.14 = 87.5 ± 12.28
Therefore, all quiz grades fall within 75.22 and 99.78, which is within 2 standard deviations of the mean. So, the correct response is 2.