Question

the mean score on a statistics exam is 84 points, with a standard deviation of 6 points. apply chebychevs theorem to the data using k = 2. interpret the results. chebychevs theorem states that the portion of any data set lying within k standard deviations (k>1) of the mean is at least 1 - 1/k². this number can be expressed as a fraction or a percentage. by the definition of chebychevs theorem, the proportion of data that lie within k = 2 standard deviations of the mean can be determined using the formula 1 - 1/(2)² = 3/4 (type an integer or a simplified fraction.) now, identify the size of the range in terms of the standard deviation. find the value that is 2 standard deviations below the mean. x - ks = 84-(2)(6)=72 next, find the value that is 2 standard deviations above the mean to complete the range. x + ks = 84+(2)(6)=96 therefore, at least □% of the exam scores fall between 72 and 96.

Explanation:

Step1: Recall Chebychev's formula

The formula for the proportion of data within \(k\) standard - deviations of the mean is \(1-\frac{1}{k^{2}}\), where \(k > 1\). Here, \(k = 2\).

Step2: Calculate the proportion

Substitute \(k = 2\) into the formula \(1-\frac{1}{k^{2}}\). We get \(1-\frac{1}{2^{2}}=1 - \frac{1}{4}=\frac{3}{4}\).

Step3: Convert to percentage

To convert the fraction \(\frac{3}{4}\) to a percentage, we use the formula \(\text{Percentage}=\frac{3}{4}\times100\%\). So, \(\frac{3}{4}\times100\% = 75\%\).

Answer:

75