a distribution of data has 80 data values, a mean of 27.8 and a standard deviation of 3.2. what can be said…

Explanation:

Step1: Calculate the number of standard - deviations from the mean

First, find the number of standard - deviations $k$ for the lower and upper bounds. For the lower bound $x_1 = 21.4$, $k_1=\frac{\mu - x_1}{\sigma}=\frac{27.8 - 21.4}{3.2}=\frac{6.4}{3.2}=2$. For the upper bound $x_2 = 34.2$, $k_2=\frac{x_2-\mu}{\sigma}=\frac{34.2 - 27.8}{3.2}=\frac{6.4}{3.2}=2$.

Step2: Apply Chebyshev's theorem

Chebyshev's theorem states that for any data set and any constant $k>1$, the proportion of data within $k$ standard - deviations of the mean is at least $1-\frac{1}{k^{2}}$. Here, $k = 2$, so the proportion of data within 2 standard - deviations of the mean is at least $1-\frac{1}{2^{2}}=1 - \frac{1}{4}=\frac{3}{4}$.

Step3: Calculate the number of data values

We have a total of $n = 80$ data values. The number of data values within 2 standard - deviations of the mean is at least $n\times(1-\frac{1}{k^{2}})$. Substitute $n = 80$ and $k = 2$ into the formula: $80\times\frac{3}{4}=60$.

Answer:

60