Chebyshev’s rule and the Empirical Rule give us information about our data set.
ID: 3208286 • Letter: C
Question
Chebyshev’s rule and the Empirical Rule give us information about our data set. Suppose you were an instructor and you gave a test to 380 students, the mean test score was 78 points and the standard deviation was 8 points. Can you answer the following? If yes, state how many and explain how you got your answer. If not, say so and explain why you cannot answer?
A. The test score distribution was symmetric and bell shaped. Can you make a prediction about how many students you would expect to score between 70 and 86 points?
B. The test score distribution's shape was not at all bell shaped. Can you make a prediction about how many students you would expect to score between 70 and 86 points?
Explanation / Answer
78-8 =70
78+8 =86
(A) Yes, using Empirical Rule
If a data set has an approximately bell-shaped relative frequency histogram, then approximately 68% of the data lie within one standard deviation of the mean, that is, in the interval with endpoints ± for populations
So, 68% of students i.e., 380*0.68 = 258.4
(B) No, we cannot predict with in 1 sigma
The Empirical Rule does not apply to all data sets, only to those that are bell-shaped, and even then is stated in terms of approximations.
With Chebyshev’s Theorem, at least 11/k^2 of the data lie within k standard deviations of the mean, that is, in the interval with endpoints ±k for populations, where k is any positive whole number that is greater than 1.