Question

the mean score on a statistics exam is 84 points, with a standard deviation of 4 points. apply chebychevs theorem to the data using k = 2. interpret the results. at least 75% of the exam scores fall between 76 and 92. (simplify your answers.)

Explanation:

Step1: Recall Chebychev's Theorem

Chebychev's Theorem states that at least $1-\frac{1}{k^{2}}$ of the data lies within $k$ standard - deviations of the mean. Given $k = 2$, then $1-\frac{1}{k^{2}}=1-\frac{1}{4}=\frac{3}{4}=75\%$.

Step2: Calculate the lower and upper bounds

The mean $\mu = 84$ and the standard deviation $\sigma=4$. The lower bound is $\mu - k\sigma=84 - 2\times4=76$. The upper bound is $\mu + k\sigma=84+2\times4 = 92$.

Step3: Interpretation

At least 75% of the exam scores fall within 2 standard - deviations of the mean. Since the mean is 84 and the standard deviation is 4, at least 75% of the scores are between 76 and 92. This means that a relatively large proportion (at least 75%) of the students' scores on the statistics exam are clustered within the interval from 76 to 92 points.

Answer:

At least 75% of the exam scores are in the interval [76, 92]. This indicates that a significant portion of the students' scores are clustered within 2 standard - deviations of the mean score of 84.