Question

the variable x is known to have a population mean of 3990, but we have no information about the shape of its distribution. at least what percentage of the data will fall within 2.95 standard deviations of the population mean? report your answer to one decimal place.

Explanation:

Step1: Recall Chebyshev's theorem

Chebyshev's theorem states that for any data set (regardless of the shape of the distribution), the proportion of data within \(k\) standard - deviations of the mean is at least \(1-\frac{1}{k^{2}}\), where \(k> 1\).

Step2: Identify the value of \(k\)

Here, \(k = 2.95\).

Step3: Calculate the proportion

Substitute \(k = 2.95\) into the formula \(1-\frac{1}{k^{2}}\). So, \(1-\frac{1}{2.95^{2}}=1 - \frac{1}{8.7025}\).
\[

$$\begin{align*} 1-\frac{1}{8.7025}&=\frac{8.7025 - 1}{8.7025}\\ &=\frac{7.7025}{8.7025}\\ &\approx0.885 \end{align*}$$

\]

Step4: Convert to percentage

Multiply the proportion by 100 to get the percentage. \(0.885\times100 = 88.5\%\)

Answer:

88.5%